*"Dinimig de Chù air Èill"*le Giacomo BALLA (1912)

**§ 3 - RO-AINILIS AIR A' CHIAD TRÌ STRUCTAIREAN-CÈILLE MODALACH.**

**__________________________**

**§ 3 - PRELIMINARY ANALYSIS OF THE FIRST THREE MODAL STRUCTURES OF MEANING.**

**§ 3 - Voorloopige analyse van de drie eerste modale zin-structuren.**

**__________________________**

**C - Brief analysis**

**of the original (mathematical) meaning of**

**MOTION**

**in its coherence with the original meanings of number and space.**

**C - Summiere analyse**

**van den originairen zin**

**der**

**BEWEGING**

**in zijn**

**samenhang met den zin**

**van getal en ruimte.**

In the modal structure of the law-sphere of movement (in its original mathematical sense intended in pure kinematics) there are very clear numerical and spatial retrocipations. Neither in the numerical, nor in the spatial aspect can we find movement in its original modal meaning of continuous flowing, which needs no further qualification.

**The differential as an anticipation of movement**

**in the original meaning of number.**

**Het differentiaal als bewegings-anticipatie**

**in den originairen zin van het getal.**

When a mathematician tries to develop, theoretically, the numerical relations between two variable magnitudes in conformity to the arithmetical laws, he makes use of the concept of function. Then one of two variables is conceived of as a function of the other (the independent variable). In this case discrete quantity is thought of as variable. But neither in the

*logical processus (the movement of thought), guiding the differential and integral calculus, nor in the differential relation between the series of values traversed by the two magnitudes, is there any question of movement in its**original modal meaning.* The differences traversed in the course of their changes by the variables -x- and -y- in the functionally coherent series of values, remain

*discrete arithmetical values.*But under the guidance of the theoretical movement of thought [1] the numerical aspect*approximates*the original continuity of pure movement in the anticipatory function of the differential quotient.______________________

[1] This guiding function of theoretical logic will be explained in the discussion of the opening-process of the modal meaning-structures.

______________________

The differential function of number expresses nothing but the limiting value of the quotient:

∆ x

∆ y

when both differences approximate zero infinitesimally.

A mathematician who is of a rationalistic frame of mind, is apt to deny any necessary connection between the differential function of the numerical meaning-aspect and the original modal meaning of movement. Perhaps he will object that the differential and integral calculus has a pure mathematical value in itself and that its relation to physics is nothing but a particular instance of its applicability. This would doubtless be correct. But it has nothing to do with the point in question.

Our statement that the numerical aspect of meaning in its infinite differential function approximates the original modal meaning-kernel of movement, naturally does not imply that movement could be taken here in the sense of an actual physical process. The word movement in this case is taken to refer to the nucleus of the modal meaning of the aspect which delimits the mathematical field of pure kinematics (phoronomy).

The logicist cannot accept the irreducible character of this modal aspect of meaning. He will try to reduce it to its 'logical origin'. The logical movement of thought will be a sufficient basis to him for the infinitesimal calculus.

**The logical movement of thought as a retrocipation**

**of the original aspect of movement.**

**De logische denkbeweging als gefundeerde retrocipatie.**

The logical movement of thought is, however, an analogical figure of meaning. It evidently refers back to its substratum in the original aspect of movement. Though remaining what it is, viz.

*logical*processus, it has a retrocipatory character and appeals to the nuclear sense of its foundation. The concepts 'variable' and 'differential' would be without any basis, if the cosmic coherence of meaning between the number-aspect and the aspect of movement in their original sense were denied.

As to movement in its original sense, it should be observed that as late as in KANT (who, at least at this point, followed in the steps of NEWTON) the prevailing view was that movement was something occurring

*in*mathematical space. This idea was due to a misinterpretation of the original meaning of movement, because it was based on the objective sensory image of space. In our psychical-sensory perception the sensory impression of movement is really found in the objective sensory image of space. The reason why this is necessarily so in accordance with the cosmic temporal order, is a subject for later research. But there can be no question of an original movement in the original meaning of space. [Van een originaire beweging in den originairen ruimte-zin kan echter geen sprake zijn.(WdW Deel 2 p72)]

**The erroneous view of classical physics concerning the relation**

**between sensory phenomena and absolute space.**

It is very important to stress this modal state of affairs, since NEWTON, led astray by the fact that physical experiments are related to objective sensory phenomena, wrongly supposed that the latter can be conceived as occurring in the 'absolute' space of mathematics. It was only a quite natural result of this lack of distinction between the different modal aspects of experience that 'matter' was viewed as a 'filling up' of this mathematical receptacle [2].

________________________

[2] The Marburg School among the neo-Kantians, too, has stuck to this opinion. NATORP in his work on the logical foundations of the exact sciences, writes with regard to the modern concept of energy: "It is exclusively the logical demand of univocal determinateness of being in relation to time and space which leads to the necessary pre-supposition of a substance of occurrence that maintains itself unchanged. This substance is something 'real', which according to its pure concept must necessarily be conceived of as always identical with itself in its fundamental existence, but as having a movable

*space-content*in space." ["So ergibt sich allein durch die logische Forderung der eindeutigen Bestimmtheit des Seins in Bezug auf Zeit und Raum die notwendige Voraussetzung einer unveränderlich sich erhaltende Substanz des Geschehens, oder eines "Realen", welches nach diesem seinen reinen Begriff notwendig zu denken ist als in seinem Grundbestand immer sich selbst identischer, dagegen im Raum beweglicher Raum*inhalt.*"]*(Die logischen Grundlagen der exakten Wissenschaften,*2e Aufl., 1921, p. 349). This statement again shows how much the Kantian form-matter scheme is prejudicial to a clear idea of meaning. KANT was already led astray by it, when he wanted to define the relation between space and moving matter.

________________________

According to NEWTON, this receptacle was conceived as a metaphysical entity: the sensorium Dei. In this metaphysical interpretation of 'absolute space' the antinomic character of the conception of sensible 'matter' as a 'filling up' of the former was sharply accentuated.

It was therefore quite understandable that KANT in his critical period transformed NEWTON's "absolute space" into a transcendental form of intuition.

But, since this transcendental form was identified with space in its original modal sense, KANT's conception remained burdened with the antinomy that sensory space is to be viewed as subjected to the purely mathematical rules of Euclidean geometry [3].

_________________________

[3] The dark schematism-chapter of the

*Critique of Pure Reason*could only*mask*this antinomy, because it does not deal with*sensory*space but with an exact Euclidean one, related to*a priori*intuition, which, as such, cannot be of a sensible nature. The question how sensible space can be subjected to the

*a priori*rules of EUCLIDEAN geometry is neither raised nor solved. The chief point is that in KANT's exclusively mathematical-physical conception of human experience there was no room for a 'sensory space' in its objective psychological meaning. For this very reason his transcendental aesthetics and his schematism-chapter could not refute HUME's psychological critique of 'exact geometry'.

Sensory perceptions as such can only be related to objective sensory space, not to an

*a**priori*mathematical one._________________________

This view, according to which 'pure Euclidean space' is an

*a priori*receptacle of sensory perceptions*("Anschauungsraum"),*had already been refuted by HUME with striking arguments. But even CARNAP maintained it in his remarkable treatise*Der Raum,*although only with respect to the topological space of intuition (not as to the metrical and projective ones, which, according to him, lack a*priori*necessity). And it is this first misconception which lies at the basis of the classical physical view that sensible movement of matter is considered as occurring in the cadre of pure mathematical space.

**Movement in its original modal sense and in its analogical meanings.**

This misconception is of a very complicated character. This appears as soon as we pay attention to the original modal sense of movement in its inter-modal relation to its analogical meanings in physics and in the psychological theory of perception.

In Aristotelian philosophy the analogical character of the fundamental concept of movement was clearly seen.

The common moment, implied in the different meanings of this concept, was found in 'change' (quantitative change, change of place, change of qualities, substantial change). But it was not overlooked that this meaning-moment was itself of an analogical nature.

The very fact that Greek thought was ruled by the dialectical form-matter motive explains its resigning to a fundamental analogy. No further inquiry was made into the original modal meaning-structure of movement to which all its analogical meanings must refer. It was in the last analysis the lack of a radical unity in the religious point of departure that prevented philosophical thought from penetrating to the original meaning-kernels of the modal aspects of human experience.

As soon as religious primacy was ascribed to the form-motive, all attention was directed to the 'substance' which must be the

*ὑπόθεσις*of every movement, the accidental as well as the substantial. But the metaphysical concept of substance could not transcend the modal diversity of meaning implied in the analogical concept of movement. The ancient Ionian philosophy of nature ascribed primacy to the religious matter-motive. Consequently it reduced all natural movement to the eternally flowing Stream of life as the divine Origin. But for this very reason this original divine movement was not conceived in an original

*modal*sense in which its modal nucleus is contained. Rather it was understood in the analogical sense of*vital*movement, which was absolutized to the divine Origin of all things appearing in an individual form and therefore subject to decay. It was only in kinematics as a branch of pure mathematics that the original modal meaning of movement could be grasped. Here movement presents itself in its modal nucleus of continuous flowing in the succession of its temporal moments. It is evident that NEWTON's well-known circumscription of 'absolute'

*or 'mathematical' time was nothing but a concept of uniform movement in this original modal sense.* It makes no sense to define the latter in the Aristotelian manner as a change of place. For movement in its original modal sense cannot be

*qualified*by spatial positions. A change of place conceived of as an intrinsic characteristic of movement would imply that movement occurs in a statical spatial continuum, and that from moment to moment it has another defined place in it. But this supposition leads theoretical thought into inescapable antinomies since it cancels the concept of movement. We shall return to these antinomies in a later context.

**The spatial analogy in the modal structure of the kinematic aspect.**

It is true that the modal meaning-kernel of movement needs an analogy of space in the modal structure of the kinematic aspect itself. But this analogy is qualified by the meaning-kernel of this aspect, not inversely. It is a

*flowing*space in the temporal succession of moments, not a statical one in the simultaneity of all its positions. This flowing space is

*founded*in the latter but cannot be identified with it. It refers indeed to the meaning-kernel of the spatial aspect, but only in the inter-modal relation of the two modal law-spheres concerned, which is guaranteed by the cosmic time-order. This spatial analogy (flowing extension) also implies an analogy of spatial dimensionality in its original sense, i.e. the*directions*of movement in flowing space, whose multiplicity in its turn is founded in the numerical aspect. It must be observed emphatically that this provisional analysis of the modal structure of movement in its original (non-analogical) meaning has nothing to do with a speculative construction inspired by a preconceived system of modal law-spheres. On the contrary, in the first (Dutch) edition of this work I tried to reduce the original sense of movement to the meaning-kernel of the modal aspect which is the specific field of physics. But it appeared later on that this attempt could not satisfy the demands of an exact analysis and must lead philosophical thought into inner antinomies.

**Physical movement as an analogy qualified by energy.**

In the first place it must be noted that in physics the concept of movement usually has a restricted application, namely in mechanics only. For this reason it might produce a confusing effect if movement is elevated to the rank of the modal nucleus of meaning of the physical aspect.

It is true that this objection cannot be decisive, because scientific terminology often lacks philosophical precision and the word 'movement' does not have an exclusively mechanical sense.

There is, however, a much more cogent argument preventing us from conceiving movement as the original meaning-kernel of the physical aspect. This is the undeniable fact that in its physical use the term movement requires a

*specific modal qualification.*Physics, in all its subdivisions, is always concerned with functions of*energy*(potential or actual) and energy implies causes and effects. That is to say that physical movement cannot reveal the original nuclear meaning of movement, but must have an analogical sense, qualified by the very meaning-moment of energy. In its original modal sense movement cannot have the meaning of an effect of energy. That is the very reason why kinematics or phoronomy can define a uniform movement without any reference to a causing force and why the physical concept of acceleration does not belong to kinematics but to physics alone. Therefore GALILEO could define the principle of inertia in a purely mathematical-kinematical way, which signified a fundamental break with the Aristotelian conception. Since movement in this original sense cannot be reduced to the numerical, the spatial or the physical aspects, it must be an original modal aspect of human experience, which is at the foundation both of physical movement and of movement in the objective psychical sense of sensory perception. That is to say that human experience of movement can never be exhausted in its objective sensory aspect. It always implicitly (in naïve experience) or explicitly (in theoretical experience) refers to the original aspect of movement which, as such, is of a pre-sensory character. We would not be able to perceive movement with the eye of sense, if this sensory perception was not founded in the original intuition of movement as an irreducible aspect of human experience. The sensualistic view is refuted by a serious analysis of the modal structure of sensory movement-perception which lays bare the analogical and referring character of the latter.

Therefore GALILEO followed the right Scientific method when he founded his mechanical theory in a mathematical kinematics. And NEWTON's conception of 'mathematical time' has not lost its scientific value if it is conceived in the original sense of pure kinematics. It is only the metaphysical absolutization of kinematic time-order and its confusion with the physical one which must be abandoned. But this does not imply that the latter may be conceived without any (at least implicit) reference to kinematic time.

Movement in its original modal sense cannot be conceived without its inter-modal reference to the original meaning of space. We would not have an intuition of a flowing extension without its inter-modal coherence with a statical space. But it is not true that this intuition needs a sensory perceptible system of reference. Only the objective sensory

*image*of movement demands the latter. But this sensory image appeals to our pure intuition of movement in its original modal meaning. It is founded in this pure intuition by the inter-modal order of cosmic time and cannot be experienced in purely sensory isolation. The sensory image of movement occurs within a sensory space of perception which itself is only an objective sensory analogy of space in its original meaning. Therefore it also appeals to the original spatial aspect of our experience. We shall return to this complicated state of affairs in a later context. The whole conception of moving matter as a filling up of space is exclusively oriented to the sensory aspect of experience. It has a psychological, not a physical or kinematic content.

Of course it is true that in physical experiments sensory perception is indispensable. But in the theoretical interpretation of the sensory phenomena the latter must be related to the modal aspect of energy which is not of a sensible nature. Fields of gravitation, electro-magnetical fields, quanta, photons, electrons, neutrons, protons, and so on, are not sensory phenomena, although the

*real events*in which they manifest themselves have all*objective sensory aspect.*They function within the original aspect of energy. But they have an inter-modal relation to the sensory aspect of human experience and in physics the objective sensory phenomena can only be theoretically interpreted as sensory*symbols*referring to the original physical states of affairs which present themselves to the physical aspect of experience.**The general theory of relativity**

**and the un-original character of physical space.**

**De algemeene relativiteitstheorie**

**en het niet-originair karakter der bewegingsruimte.**

The general theory of relativity has made the discovery that the properties of physical space (i.e. essentially energy-space) are really determined by

*matter*(in its physical function of energy), because of the indissoluble coherence of physical space and physical time. This is the reason why no privileged rigid system of co-ordinates for physical movement can be accepted [4]._________________________

[4] The general theory of relativity utilizes the so-called Gaussian co-ordinates, i.e. the four-dimensional (including physical time as the fourth co-ordinate) system of co-ordinates with curves varying from point to point. They can only be understood as physical anticipations in geometry, in so far as this geometrical pattern is related to physical states of affairs.

_________________________

If the properties of physical space depend on energy, the analogical character of this space is indisputable. The general theory of relativity, in the nature of the case, is unable to conceive of physical space without its inter-modal coherence with original space, in so far as the latter anticipates the meaning of energy. Such an anticipation necessarily makes an appeal to the original meaning of energy. Hence it can be admitted that the geometrical foundations of the general theory of relativity (in the transcendental direction of time) are dependent on the modal meaning of energy.

EINSTEIN formulates this as follows: 'According to the general theory of relativity the geometrical properties of space are not independent, but they are determined by matter' [5].

_________________________

[5]

*Über die spezielle und die allgemeine Relativitätstheorie*(12. Aufl.), p. 76: "Gemäss der allgemeinen Relativitätstheorie sind die geometrischen Eigenschaften des Raumes nicht selbständig, sondern durch die Materie bedingt."_________________________

But this statement can only be correct, if 'matter' is not intended as a filling-up of original space but rather in its physical function as qualifying its own extension. The question whether this analogical space is a

*continuum*cannot be answered in an*a priori*way. It is well known that by accepting the classical view of the continuous character of physical space the theory of relativity does not completely agree with the modern quantum-theory of energy [6]._________________________

[6] Particularly the famous French physicist DE BROGLIE has discussed the philosophical problems implied in this incongruence.

_________________________

In the theory of the modal law-spheres there would be no single difficulty in abandoning this residue of the classical conception. For the analogical character of physical space and

*its qualification by the meaning-kernel of the energy-aspect is here clearly seen.* If the energy-aspect in its factual side appears to have discontinuity, it is quite understandable that physical space is determined by this discontinuous structure.

Only a theoretical view of reality which lacks a clear distinction between the modal aspects of human experience and holds to the Kantian view of Euclidean space as an

*a priori*form of sensory intuition, must reject the conception of a discontinuous space as paradoxical. If the modal boundaries of meaning between original space and its kinematical, physical and sensory analogies are obliterated, there arises indeed an inner antinomy. That is to say, an antinomy arises if it is assumed that the structure of space is dependent on a matter which itself is 'enclosed in pure space', consequently, which itself must be determined by the pure mathematical properties of the latter.

**The discretion of spatial positions and the un-original or analogical character of this discretion.**

**De discreetheid der ruimtelijke posities en het nietoriginair, maar analogisch karakter dezer discreetheid.**

In the original meaning of space the positions of the figures must necessarily retain their discretion in the modal continuity of their extension. This discretion, as an arithmetical analogy, is founded in the original meaning of discrete quantity. It is indeed no original kind of discretion. The discrete magnitude, e.g., of the three sides of a triangle, depends on points that have no actual subjective existence in space themselves, as they have no extension in any dimension.

This discretion is to be understood in the static sense of the original spatial positions, which cannot flow into one another in the original meaning of motion. The totality of the spatial positions, passed through by a point, a line, a plane, merely in

*imagination,*in the mathematical*movement of thought,*is not subjectively actual in the original*spatial aspect of time.*No more is the totality of the finite numbers in an approximative series subjectively actual in the modal meaning of*arithmetic**time.* The original time of the spatial aspect is one of the modal meaning-functions of cosmic time, whereas cosmic time itself has an inter-modal continuity. In space the meaning of time is spatial

*simultaneity*[7]*,*not that of*kinematic succession.*__________________________

[7] Also PLATO in his dialogue

*Parmenides*has stressed the fact that spatial simultaneity is a real modus of time.*__________________________*

But in the idea of the totality of the discrete positions of a spatial figure conceived of as being subject to 'continuous transformation', original spatial time approximates the meaning of kinematic time, in so far as it anticipates the meaning of kinematic succession.

**The antinomies of ZENO are due to the attempt to reduce the modal meaning of motion to that of space.**

**De antinomieën van Zeno ontstaan door de poging, den modalen zin der beweging tot dien der ruimte te herleiden.**

No attempt should be made to reduce succession in the original meaning of

*motion*to the discrete simultaneity of an infinite series of magnitudes in the original meaning of*space.*For then theoretical thought will inevitably be entangled in the notorious antinomies, already formulated by ZENO the Eleatic (ACHILLES and the tortoise; the flying arrow). His dialectical arguments against the possibility of movement could only show that movement can never be construed from an approximative infinite series of discrete spatial magnitudes. From these antinomies it is at the same time clear, that the opposite procedure is equally impossible: discrete spatial magnitudes cannot flow into one another in the continuous succession of movement.

CASSIRER makes the remark that geometry has developed a rigorously systematic treatment of its province and has devised truly universal methods only after changing over from the geometry of measure to the geometry of spatial positions [8].

__________________________

[8]

*Substanzbegriff und Funktionsbegriff,*p. 99/100.__________________________

This development, following LEIBNIZ' programme of an

*analysis situs,*resulted in the theoretical opening of the modal functions of the spatial aspect that anticipate the original meaning of the aspect of motion. But this is bound to the condition that theoretical thought does not attempt to violate the sphere-sovereignty of the modal aspects.**Analytic and projective geometry viewed in the light**

**of the theory of the law-spheres.**

**Analytische en projectieve geometrie in het licht**

**van de theorie der wetskringen.**

In DESCARTES' analytic geometry the spatial series of positions anticipating the original meaning of the aspect of motion are not really analyzed in the modal meaning of space, but replaced by the anticipatory functions of number. The different spatial

*forms of the plane curves are conceived as proceeding from the 'movement' of a definite point, fixed as their fundamental element. Its position in space has been determined univocally by means of a system of co-ordinates. The points obtained in this way are approximated from the values of the numbers assigned to them.* LEIBNIZ' programme of an

*'analysis situs'*was primarily intended to discover the anticipatory principle of progression in the aspect of space itself. This programme was essentially carried out in PONCELET's founding of projective geometry [9].__________________________

[9] S. PONCELET,

*Traité des propriétés de figures*(2ième ed. Paris 1865).__________________________

In the theory of the law-spheres PONCELET's projective geometry is only to be understood as a theoretical attempt to discover the constant correlative functions of spatial figures of the same group that

*approximate*the original*meaning of motion*in an infinitesimal series of positional variations. A definite spatial figure is considered to be correlated to another if it can be derived from the other by 'a continuous transformation' of one or more of its positional elements in space.

In this process certain spatial basic relations are pre-supposed as the

*invariants*of the whole system of spatial relations. The most important form of correlation, connecting different spatial figures with one another, is discovered in the projective method. Here geometry has the task of discovering those 'metrical' and 'descriptive' moments of a figure that remain unaltered in its projection. Accordingly projective geometry now introduces the

*imaginary*spatial figure, and speaks of the*imaginary points of intersection*in the transformed system. One thing is at once clear: it must be the subjective spatial limiting functions that we are confronted with in this procedure.

This is the same thing that has been found in the imaginary functions of number, which also appeared to be subjective limiting functions.

It was owing to the discovery of these anticipatory spatial limiting functions that the principle of progression was found to establish the functional coherence between spatial systems which are otherwise entirely heterogeneous. It was seen that the invariant, positional relations in conformity to the spatial laws also obtain among the infinite series of discrete positions whose mutual positional difference is 'infinitesimally small'.

Consider, e.g., two circles in a plane. If they intersect, a common chord has been given connecting the two points of intersection. The points of this straight line are such that the tangents that can be construed from these points to the circles are equal.

This spatial relation also obtains in case the extreme limit is reached in the series of the positional changes of the two circles, i.e. when they do not intersect any longer. In this case, too, there is always a straight line — the so-called radical axis of the two circles — possessing the spatial property mentioned above and connecting the two 'imaginary' points of intersection.

In the same way it can be proved, e.g., that when three circles are given in a plane, and we construe the 'radical axes' for any two of them until they have all been used, the three lines obtained in this way intersect at one point. According to the principle of the invariant relations in the infinite series of positions, the same thing holds good for the special case that the three circles intersect indeed, etc. [10].

_________________________

[10] Cf. HANKEL,

*Die Elemente der projektivischen Geometrie*(Leipzig), p. 7 ff._________________________

On the ground of the same principle of progression the projective view of Euclidean space is entitled to speak of the infinitely distant point in which two parallel lines intersect; or of the infinitely distant straight lines in which two parallel planes intersect.

In the 'imaginary' positional functions the original meaning of space indeed approximates that of movement. Projective geometry only violates the specific modal sovereignty of the law-spheres of space and movement, in the further development given to it, e.g., by CAYLEY and KLEIN. In their theory conclusions are drawn from the principle of the invariant relations to the effect that an actual continuity is assumed in the series of the transformations of the spatial positions. In other words, they speak of an actual 'all-ness' (totality) of the changing positions in this series. This conception implies inescapable antinomies.

For in the spatial order of time this totality can no more be actually given than in the numerical order the totality of the numbers in an approximative series. The

*differential*and the*integral*of the series can no longer have original spatial meaning if the latter is considered to he actually continuous. Only in the original modal meaning-aspect of movement can there be any question of an*actual continuity*of the changes of position. But*in the meaning of original movement there are no really discrete spatial positions.* When theoretical thought tries to conceive the transition of the spatial positions in the series as 'actually closed', or 'continuous' (the pseudo-concept of a 'totality of transformations which is dense in every direction'), it again gets involved in the antinomy of 'actual infinitude'. A real continuity in the transformations would cancel the original meaning of space; but a real reduction of original movement to an infinite series of discrete spatial positions cancels the original meaning of movement.

**The logicistic shiftings of meaning in projective geometry.**

**De logicistische zin-verschuivingen in de projectieve geometrie.**

The logicistical eradication of the modal boundaries between space and movement must be understood as an unwarranted shifting of meaning. The original sense of movement is then identified with the analogical movement of thought which is actually operative in the analysis of the spatial positions.

According to F. KLEIN all the geometrical transformations resulting from the arbitrary movements of the elements in an ordinary three-dimensional space, form a group [11].

___________________________

[11]

*Einleitung in die höhere Geometrie.*II.*S. 1 ff.*___________________________

The 'movement' intended here, which overarches the entire series of positions of the 'group', is in fact the

*theoretical movement of thought.*This thought conceives the original meaning of space in its anticipatory coherence with the original sense of movement. This complicated state of things is given a perfectly erroneous interpretation, if it is suggested that the original modal meaning of the static relations of space can be dissolved into a group of

*"Operationen"*(*=*operations) in the sense of*movements of thought.* In mathematics there is a logicistic tendency which poses the dilemma:

One must either acknowledge the purely logical origin of mathematical concepts, — or fall back into the view of space as it is given in sensory experience.

But in this dilemma the

*cosmological problem of meaning*implied in the mathematical concepts, has been obscured fundamentally and essentially.**Herman Dooyeweerd,**

**New Critique of Theoretical Thought****, Vol II/ Part I/ Chapt 2/§3 pp 93-106)**

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