DIMENSIONALITY
Arbitrary Character of Dimensionality--Various Definitions of
Dimension--Real Space and Geometric Space Differentiated--The
Finity of Space--Difference Between the Purely Formal and the
Actual--Space as Dynamic Appearance--The _A Priori_ and the _A
Posteriori_ as Defined by PAUL CARUS.
In previous chapters we have traced the growth and development of the
non-Euclidean geometry showing that the so-called fourth dimension is
an aspect thereof. It is now deemed fitting that we should enter into
a more detailed study of the question of dimensionality with a view to
examining some of the difficulties which encompass it.
The question of dimension is as old as geometry itself. Without it
geometric conclusions are void and meaningless. Yet the conception
of dimensionality itself is purely conventional. 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 space and arbitrarily assign three elements to
determine its position. The simplest standard for this purpose is the
cube having three of its edges terminating at one of its corners.
D
+
A |
\ |
\ |
\ |
\ |
\|
+-------------------------
B C
FIG. 6.
Thus because it is found that the entire volume of a cube is actually
comprehended within the directions indicated by the lines _ab_, _bc_
and _db_ it is determined that the three coördinates of the point _b_
are necessary and sufficient to establish the dimensions of the cube
and consequently of the space in which it rests. The conception may be
stated in this way: If a collection of elements, say points or lines,
be of such a nature or order that it is sufficient to know a certain
definite number of facts about it in order to be able to distinguish
every one of the elements from all the others, then the assemblage or
collection of elements is said to be of the same number of dimensions
as there are elements necessary to its determination. In the above
figure there are three elements, namely, the lines _ab_, _bc_, and
_db_, which are necessary and sufficient for the determination of 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 obviated. For if it
be true that three and only three elements are necessary to determine
a point-position in our space and that in this determination we also
find the number of dimensions of space, then it may also be true that
_n_-coördinates would just as truly determine the dimensionality of
an _n_-space, which is granted. But then the _n_-space would be just
as legitimate as the three-space; for it is determined by exactly
the same standards. It is both quantitatively and qualitatively the
same. If, however, on account of the exigencies that might arise,
we are forced to seek solace in the notion of an _n_-space whither
shall we turn for it? It cannot be found; for it is imperceptible,
uninhabitable, non-existent, and therefore, absolutely and purely an
abstraction. Consequently, there must be something radically wrong with
the definition of space or with its determinants.
The purely arbitrary character of dimensionality is very aptly
described by CASSIUS JACKSON KEYSER, who says:
"... The dimensionality of a given space is not unique, but
depends upon the choice of the geometric entity for primary or
generating element. A space being given, its dimensionality
is not therewith determined, but depends upon the will of the
investigator who by a proper choice of generating element endows
the space with any dimensionality he pleases. That fact is of
cardinal significance for science and philosophy."[11]
[11] Vide _Monist_, Vol. XVI, 1896, Mathematical
Emancipations.
It is a fact of "cardinal significance" for science; because it
emphasizes the necessity for some more rational procedure than that
of the geometrician in arriving at an absolutely unique method 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. The
definition of perceptual space should be such as rigorously inhibits
its inclusion as a particular in any general class. The necessity for
this is warranted by its universality and uniqueness.
The lines of demarkation between what is recognized as perceptual
space and what has been called geometric or conceptual space should
be very sharply 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, properly speaking. It is merely a system of
space-measurement. And as such has no logical right to be put in the
same category as perceptual space.
Real space is unique. Geometric space belongs to a class whose members
are capable of indefinite multiplication. It is certainly most
illogical to identify them. Perceptual space, figuratively speaking,
is a quantity; analytic space is the foot-rule, the yard-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 illegitimate it is rather surprising
that an error of this nature should have been made. It is perhaps
accountable for on the grounds of the geometer's complete _insouciance_
as to how his postulates shall stand in their relation to things in the
phenomenal world.
It is agreed that as convenient as is Euclid's system of
space-measurement it is not by any means congruent with the extension
of real space objects. It does, however, approximate congruity with
these objects as nearly as possible. How then could it be expected
that a system of space-measurement 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 distinction
between their conception of space itself and its qualitative
peculiarities. They use the terms interchangeably. So that dimension
means space and _vice versa_. In this lack of discrimination may
be found the source of much of the confusion which attaches to the
conception of space.
If it were arguable that the relation between space and its dimensions
is the same as that between matter and its properties then the
restriction of this relation to three and only three directions of
extent would be disallowed; for the reason that 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 the meaning of the term dimension so that
it not only includes the idea of direction but an entirely new class of
qualities--the fourth dimension. And despite this reformation of the
original conception, they demand that it shall be called space.
We have just shown that the generic concept of dimensionality is that
three and only three coördinates are necessary and sufficient for
its determination. Granting that this is true, are we not compelled
consequently to see that we 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 explicitly admitted that the
three dimensions have a fourth. Aside from the logical difficulties
which beset these conclusions there is also set up a condition which is
at variance with the most elementary requirements of common sense.
Thus far mathematical thought has not served to clarify our notions
of space nor to shed any new light upon the vital processes which are
alleged to have their explanation in the new discovery. Simply stated,
metageometricians have brought us to the place where we must either
recognize that the fourth dimension is another sphere lying dangerously
near the earth in which space extends in four primary directions and
in which four coördinates 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 as having an unlimited number of dimensions.
To accept the latter view is equivalent to saying that, in the above
figure, the three lines _ab_, _bc_ and _db_ have formed a triple
_entente_ by which they have mutually and severally acquired a new
domain, hyperspace, and in which, because of the vast resources of the
region, they are able to perform wondrous things.
Let us examine briefly the various current definitions of dimension.
It is assumed by not a few that dimension is the same as _direction_.
But can we grant this wholly to be true? If so, then a mere child
may see that there are and must necessarily be as many dimensions 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 dimensions than has
heretofore been allowed. For the identification of the term dimension
with direction already makes available an almost infinite number of
dimensions. But this view is objected to by the advocates, for it is
contrary to the hypothesis of _n_-dimensionality.
Dimension also means _extent_. This is partially true. It cannot be
wholly true. For, if it were, then, space would have only one dimension
which is also not allowable under the hypothesis. Then the 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 possible directions
and not in any one direction. So that it is not permissible to say
that space extends in this direction or that because it extends in all
directions simultaneously and equally.
Geometers claim that space is a system of coördinates necessary for
the establishment of a point-position in it. This view, however,
identifies space with a system of space-measurement and is therefore
faulty. According to this view there may be as many spaces as 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 would be confusing. Much, if not
all, of such a system's utility and convenience would be unavailable
or useless. That, too, would be in violation of the avowed purpose of
these investigations which is to enhance the utility and convenience of
mathematic operations.
Now it is evident that space is neither direction, extension, a system
of space-measurement nor a system of manifolds whose dimensions are
generable. And this is so for the same reason that a piece of cloth
is not the elements of measurement--inches, feet, yards--by which it
is apportioned. And because we find that the fabric of space lends
itself accommodatingly to our conventional norms of measurement is not
sufficient reason 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 these purposes space is extremely elastic
and accommodates itself to the shape and scope of any construction
we may decide to try upon it. In this respect it is like water which
has no regard for the shape, size or kind of vessel into which it may
be posited. There is one thing certain that judging from the above
considerations there has been not yet any absolute, all-satisfying
definition devised for space by mathematicians.
The best definitions hitherto constructed are purely artificial and
arbitrary determinations. It is rather anomalous that there should be
so little unanimity about what is the most fundamental consideration
of mathematical conclusions which are supposed to be so certain, so
necessary and universal as to be incontrovertible. Confessedly, it
is a condition which raises again 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 of our
conception of space? It seems that we shall have necessarily, on
account of the recognized limitations of mathematics in this matter,
to turn to some more tenable source for the norms of our knowledge
concerning space. For in the light of the rather indefensible position
which metageometricians have involved themselves there appears to be no
hope in this direction.
It is undoubtedly safer not to rely altogether upon the purely
abstract, even in the world of mathesis, for any absolute criterion
of knowledge. It is perhaps well that we should expunge the word
_absolute_ from our vocabularies. It is really a misnomer and has no
meaning in the lexicon of nature. There is in reality no _absolute_ in
the sense of final absolution from all conditions or restrictions.
In the ultimate analysis there is unquestionably no hue, tone, quality,
condition nor any imaginable posture of life, being or manifestation
that is absolved from every other one of its class or from the
totality. All these are relational and interdependent. There is no room
for the absolute. In fact, it is a quality which cannot in any way
be ascribed to any aspect of kosmic manifestation. It has existence
only in the mind and has been devised for the purpose of marking the
limits of its scope. All being is relative; all life is relative and is
destined to change 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 committed by pushing the validity of these limits
to a position without the sphere of limitations. This seems to be the
crux of the whole matter. Mathematicians, notably non-Euclideans,
have sought to extend the comparatively small sphere of limits of
congruence between mathematic and perceptual space to such an extent
as to cause it to encroach upon forbidden territory. In doing this they
have erred grievously, causing serious offense to the more sensitive
spirit of the high-caste mathematicians among whom are none more truly
conservative than PAUL CARUS,[12] who says:
"Metageometricians are a hot-headed race and display sometimes all
the characteristics of sectarian fanatics. To them it is quite
clear there may be two straight lines through one and the same
point which do not coincide and yet are both parallel to a third
line."
[12] Vide _Monist_, Vol. XIX, p. 402 (1909).
To the student who has carefully followed the development of the
non-Euclidean geometry and the notion of hyperspace the above
characterization is none too severe nor ill-deserved. Nothing could
more vividly yet correctly portray the impious tactics of the
metageometrician and establish his perceptual obliquity more surely
than the mere fact, mentioned 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 acceptance of his offerings as many
of the most serious minded mathematicians are inclined to give. Is
it to be wondered at that, despite the profuse protestations of the
advocates, many who take up the study of the question of hyperspace
should experience a deep revulsion from the posture assumed by
metageometricians with respect to these queries?
Linked with the idea of dimensionality is the notion that space is
infinite. This is a conception which has its roots imbedded in the
depths of antiquity. Primitive man, looking up into the heavens at
what appeared to him as a never ending extension, was awed by its
vastness; but the minds of the most learned of the present-day men
are not free from this innate dread of infinity. It permeates the
thought life of all alike and none seems to be able to rise above it.
Mathematicians, philosophers, scientists all share in the general
belief that space is without limit, 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 is, when all the
nonsensical effusions have been cleared away from our space conceptions
and men come really to understand something of the essential nature
of space this new era will find its true beginnings in the mind of
RIEMANN. Although it must be said, as is the case with all progressive
movements, the later development of a rationale for this conclusion
will vary greatly from his original conception. For he had in mind a
space that is generable and therefore a logical construction while
ultimately the mind will swing back to a consideration of real space.
Already men are beginning to see a new light. Already they are
beginning to take a new view of space in general. The departure is
especially noticeable in the attitude assumed by HIRAM M. STANLEY.[13]
He says:
"If we seek the most satisfactory understanding of space we shall
look neither to mathematics nor Psychology but to Physics. The
trend of Physics, say with such a representative as OSTWALD, is
to make things the expression of force; the constitution and
appearance of things are determined by dynamism; and we may best
interpret space as a mode of this dynamic appearance."
Space, as a mode of dynamic appearance is a slight improvement upon the
old idea of a pure vacuity; for in the light of what we now know about
space content much of the dignity of that view has been lost. Men now
know that space is 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, life, light and
psychographic ethers. But it does not stop here. It postulates a series
of grades of finer matter than the physical which fills space and
permeates its entire extent even to identification with its essential
nature.
[13] _Philosophical Review_, Vol. VII (1898).
STANLEY continues:
"Everything does not, as commonly conceived, fall into some
pre-existent space convenient for it; but everything makes its
own spaciousness by its own defensive and offensive force, and the
totality of all appearance is space in general."
According to STANLEY, not only do physical, perceptual objects, by
their "offensive and defensive force" make their own space but the
appearance of that in which no physical object is makes room for itself
by its own dynamic force. In other words, that which we call "pure
extensity" is by virtue of its dynamism the cause of its own existence.
At first hand there appears to be little worthy of serious
consideration in this view of STANLEY; yet, if carried to its logical
conclusion, the merit of the hypothesis becomes apparent. Accordingly,
interstellar distances which are commonly said to be even without air
or life of any kind are really an appearance possessed of a dynamism
peculiar to itself. 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 perceptible to our senses. Logically, if
the dynamism were removed there would remain neither space nor the
appearance of space. If this were true, and it is worthy of serious
thought, then space is certainly finite, as in its totality, according
to STANLEY'S view, it would have to be regarded as a "phenomenon of the
inner and finite life of the infinite."
It is believed that we may go a step further and unqualifiedly assert
that _space is finite_, even denying its infinity as a "general mode
of the activity of the whole." Yet it is transfinite in the sense
that it 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 limited always to become manifest. The Deity has limited His
being in order that there may be a manifested universe. All things,
all appearances are finite; because they are phenomena connected with
manifestation.
This question may be viewed from another standpoint. All things in
manifestation or existence are polar in their constitution. For
instance: there cannot be a "here" without a "there." There cannot
be an "upper" without a "lower." Right is copolar 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? Space is paired with a lack of space. There cannot
be what we call _space_ without there being at the same time the
possibility, at least, of the _lack of space or spacelessness_. This is
a conclusion that is rigorously logical and incontrovertible.
But it has been urged that it is impossible for the mind to imagine
a condition where there is no space. It even has been asserted that
it is contrary to the constitution of the mind itself to imagine "no
space." But whether imaginable or not has no effect whatever upon the
validity of the conception. Neither, it is said, can we imagine a
fourth dimension but the mind has 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 is
no description at all or it is a true delineation of an idealized
construction which is well within the mind's powers of imagination.
Indeed the question of imaginability is not determinative in itself;
for what the mind may now be unable to imagine, because of its more or
less nebulous character, and owing to its infancy may in the course of
time be easily accomplished.
The universe is a compacted _plenum_. It is chock-full of mind, of
life, of energy and matter. These four are basically one. They exist,
of course, in varying degrees of tenuity and intensity and answer to a
wide range of vibrations. Together, 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 in which no
one of the appearances which we now perceive would be possible. Even
sheer extensity would be non-existent. All scope of motility would be
lacking. Dimension, coördinates, direction, space-relations--all would
be impossible.
A straight line is an ideal construction of the mind. It does not exist
in nature. It can never be actualized in the phenomenal universe.
Between the ideal and the real, or actual, there is a kosmic chasm. It
broadens or narrows according as the phenomenal appearance approaches
or recedes from the ideal. What, therefore, can be postulated of
the one will not apply with equal force to the other. They are not
congruent and can never be in the actualized universe. The moment the
actual becomes identified with the ideal it ceases to be the actual.
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
evolution. The phenomenal may approach the ideal as a limit, but can
never become fully congruent with it. _The difference between the ideal
and the actual is a dynamic one_; it is by virtue of this difference
that the universe is held in manifestation. Evolution is the decrement
of this difference between the purely formal and the actual. So long
then as a kosmic differential is maintained the phenomenal continues
to be manifest: when it is finally reduced to nothing it goes out of
manifestation. The phenomenal is finite; the ideal infinite.
Wherefore, it is undoubtedly improper to refer to space as being
infinite. The term really is inapplicable. Transfinity is much better
and more accurate. Space is transfinite because its scope is greater
than any finite scope of motility can encompass, because it exceeds
finite comprehensibility.
RIEMANN'S notion that space is limited gains weight in the light of the
foregoing considerations. But he could not conceive of the limitability
and unboundedness of space as such in its pure essence; 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 proceeded upon that basis.
And of course, his view while it had no reference to perceptual space
nevertheless possessed an illative relation thereto and should be
recognized as construable in that light.
The process of squaring the circle recognized as a geometric
impossibility is significant of the fluxional nature of the universal
residuum perpetually maintained between the archetypal and the
manifested kosmos. It seems that there is a profound truth embodied in
this problem. There is a lesson that may be learned by mathematicians,
philosophers, scientists and thinkers in general. There is an element
of eternal necessity and universality about it 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 circumference of a circle might be constructed, such a figure
would not be a square but a circle. The manifested universe is like
that--the process of inscribing squares within a circle. It is ever
_becoming_, _evolving_, _developing_, but never quite attains. Infinity
is a process. But no single stage in that process is infinite. Each
is finite and their totality makes the infinity of the process. The
universe manifested to the senses or the intellect is finite.
"Space," says PAUL CARUS, "is the possibility of motion in all
directions."[14] To be sure, it is admitted that space offers
opportunity for motion in all directions. But is space this opportunity
of motility? Or is possibility of motion space? The possibility of
motion must rest in the thing that moves. It implies a potency in the
moving entity, not in space. If it is meant that space is the potency
that resides in the moving 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 construction. It cannot
be found in nature. Pure form is _ideal_. Impure or natural form is
actual. Therefore the space in which we live and in which the universe
exists cannot be a "pure form" because life cannot exist in the purely
formal. It is useless to talk about space as mere form so long as
it maintains life. The difficulty which this phase of the question
presents is another evidence of the inadequacy of our definitions.
[14] Vide _Foundations of Mathematics_, p. 107.
It is also found to be impossible to concur in CARUS' conception of
knowledge _a priori_. His notion of the _a priori_ varies somewhat from
the Kantian view. He defines it as an "idealized construction," the
"mind made," "abstract thought," and places it in the same category as
a concept. This is undoubtedly born of his desire to get rid of KANT'S
"innate ideas" which seem to be distasteful to him. But in doing so it
appears that the real _a priori_ has been overlooked. Let us examine
for a moment this important question. The _a posteriori_ connotates all
knowledge gained through the senses, 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,
comparing and judging. Almost the entire range of thought, its entire
scope, is characterized by the _a posterioristic method_. In fact,
all thought is _a posterioristic_. Despite the fact that, in thinking
in the abstract, it is necessary mentally to remove all elements of
concreteness, all materiality and all actuality, the conclusions
reached have to be referred to the standards maintained by the actual,
the concrete and the material. Then we do not start with the abstract
in our thinking. We begin with the concrete and by mentally removing
all physical qualities arrive at the abstract.
The mind has a constitution. It acts in a given way because it is its
nature so to act. Not because it has learned to act in that manner. It
performs certain functions intuitively without previous instruction
or experience for the same reason that water dampens or heat warms.
It is natural for it to do so. This naturalness, this performance of
function without being taught or without experience constitute 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 mental attitudes of
affirmation, negation or indifference because we have learned to do so
by virtue of the tuitional capability of mind. These describe the _a
posteriori_. That is, all knowledge obtained as a result of voluntary
mental processes constitutes the mass of knowledge _a posteriori_. The
_a priori_ is what the mind is by nature: the _a posteriori_ is what
the mind becomes. It is the mind-content.
The _a priori_ is not a mental construction; it is an essential
principle of mind. It should not be identified with the "purely
formal," as is done by PAUL CARUS:[15]
[15] Vide _Foundations of Mathematics_, p. 42.
He says:
"The _a priori_ is identical with the purely formal which
originates in our mind by abstraction. When we limit our attention
to the purely relational, dropping all other features out of
sight, we 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
advantage of being applicable to any purely formal consideration
and we work out systems of numbers which, when counting, we can
use as standards of reference for our experience in practical
life."
Thus CARUS definitely links up the _a priori_ to a factor which is
nothing more nor less than a mental by-product. For such is the
category in which would be placed both the process of abstraction and
its results. It is therefore exceedingly difficult to understand why
so cursory a consideration should have been given to the principle of
_apriority_ than which no other element of mind is more essentially a
part of the mind itself.
The formal is symbolic. It signifies an informing quantity. Pure form
itself is but a negation of that which formerly filled it. Then, too,
the formal is purely artificial because it is a mental construction.
Essentially there is as much difference between the purely formal and
the _a priori_ as between creator and creature, as between potter and
clay. The one is the builder, the other is the material; 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 the number of possibilities
open to the mind is, on account of its nature and purpose limited, it
is not to be supposed that it (the mind) shall measure up to all the
possibilities offered by the formal. Moreover, it is certain that no
sane mind cherishes the hope that there shall ever be found in the
universe of life and form a congruence for all of the possibilities
held out by the purely formal.
As an eternal principle of mind, the _a priori_ is in agreement with
the divine mind of the kosmos. In its _aposteriority_ the mind is of
diverse tendences, qualities and characteristics. Apriorily, it acts in
unison with the eternal purpose of life and the universal mind. In its
aposteriority, it often goes awry. In its _apriority_ it can never be
insane; insanity is a symptom of the morbid _a posteriori_.
The mind in man acts the same as mind in the vegetal and lower animal
kingdoms. Metabolism and katabolism, indeed all cell-activity, are
_a priori_ performances of the mind. Growth and all its phenomena,
the cyclicism of natural 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 performed without any assistance from the
intellect. All the myriad activities in nature with which the intellect
in man has not the slightest concern, truly acting in accord with some
primordial impetus, are activities _a priori_.
Now what is the attitude of the intellect, in the light of the _a
priori_, towards space and the question of dimensionality? It is
evident that no matter what this attitude may be it is in agreement
with the constitution 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 liable to the illusionary drawbacks common to
conclusions based upon limited experience. The geometric view of space
belongs to the category of the _a posteriori_. Hence it is subject to
the usual imposition of error.
Tersely stated, KANT's view of space is that it is a form of intuition,
a form _a priori_, a transcendental form. As such he considered it
to be a native form of perception not belonging to the category of
sense-deliveries. Accordingly, space is a form of intuition arising
out of and inhering in the constitution of mind. It is a notion
which constitutes the universal and eternal prerequisite of mind and
is, therefore, intrinsically necessary to all phases of mentation.
Now, this being true just what may be said to be the relation of
dimensionality to this _a priori_ form of space 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 coördinates for final determination? There is no other
answer but that the mind makes no such reference and is dependent upon
no kind of coördinate system in any of its operations _a priori_. As a
form of intuition, the space notion is present in the mind as a scope
of existence, of motility, of being and of sheer roominess. The notion
of direction or dimension, being an artificial construction, does not
enter into this form of intuition at all. It is only when the mind
comes to elaborate upon its perceptive performances and possibilities
that the questions of relations, positions and directions arise.
But this latter is a matter separate and distinct from the state of
awareness which embodies the notion of space.
Dimension is an arbitrary norm constructed by the mind for the
determination of various positions in space. It is an accident or
by-product of the process of elaborative cognition, a convenient
and appropriate means of measurement for objects in space and their
space-relations. But it is no more _a priori_ than a foot rule or a
square. But being purely an empirical product it may be said to be
an aspect of psychogenesis because it relates to the evolutionary
aspect of mind. The assumption may therefore be allowed that the
mind may, in the course of its evolution, find it convenient 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 the same; there has been merely a
complication of environmental influences. Similarly, in the acquisition
of four-dimensional powers, granting that such an acquisition is
possible, there is nothing to be added to the _aprioriness_ of mind
_itself_. Is it not, therefore, logical to assume that the discovery
of a fourth coördinate and the consequent conceptualization of the
same, point to the development in the mind of a greatly extended
faculty, more keenly penetrative powers of cognition and a further
diversification of its environments than it has hitherto enjoyed?
Indeed, it seems so.