"Sealladh-tìre le Tuiteam Icarais" le Pieter Breugel (1558)
§ 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.
B - Brief analysis
of the original modal meaning of
in its coherence with the meaning of number.
of the original modal meaning of
in its coherence with the meaning of number.
B - Summiere analyse
van den originairen modalen zin der
in zijn samenhang met den zin van het getal.
van den originairen modalen zin der
in zijn samenhang met den zin van het getal.
The structure of the original modal meaning of number does not show any retrocipation. Original quantity does not have modal substrata. According to their modal structure of meaning all the other law-spheres are founded in the numerical aspect. This means that the latter is the first modal terminal sphere of our cosmos.
MEINONG's 'Gegenstandstheorie' and G. H. T. MALAN's
critique of the first modal law-sphere.
This will be denied by Aristotelian scholasticism, which holds to the view that the 'ontological category' πόσον (how much?) pre-supposes numerable 'matter' in its spatial extension.
But this metaphysical view is not founded on a real analysis of the modal structures of the different aspects of human experience. The analysis of the modal structure of the spatial aspect will demonstrate that the latter pre-supposes the numerical one.
From a quite different standpoint my view of the numerical aspect as the first terminal aspect of human experience has been attacked by G. H. T. MALAN, emeritus professor of philosophy at the University of the Oranje Free State (S. Africa), in his treatise The First Sphere of DOOYEWEERD (Die Eerste (Getals-) Kring van DOOYEWEERD), published in the Tijdskrif vir Wetenskap en Kuns of the S. African Academy of Sciences and Arts (Oct. 1949), p.101 ff. This author starts from the so-called "Gegenstandstheorie" of A. MEINONG and is of the opinion that the numerical aspect pre-supposes pre-numerical sets of discrete objects which are sensorily perceptible, e.g., a pair of shoes, twins, and so on.
He also interprets RUSSELL's class-concept "gegenstandstheoretisch" in this sense, although he agrees that RUSSELL himself has conceived of the concept of class (an 'incomplete symbol') as a purely logical notion.
The chief objection raised by him against my conception of the meaning-kernel of the numerical aspect is that I have failed to indicate the original objects which have the quantitative mode of being:
"The objects which have number lie in altogether different spheres. They are points, stones, apples, movements and so on. But none of them belong to the first (i.e. the numerical) sphere. DOOYEWEERD is not aware of this lack of specific substantial objects in the sphere. Nevertheless, he speaks about the latter as if there are such objects and calls them 'numbers'. What kind of objects can these numbers be, and from where does he get them? The answer is: he constructs them in a metaphysical way. He postulates first a mode of being or modal meaning, i.e. quantitative discreteness in abstracto. Then he hypostatizes this mode of being or meaning and gets his entity 'number'. 'Number' as an object is the hypostatized quantitative mode of being. From the mode of being itself 'number' is born."
This whole manner of criticism testifies to the fact that MALAN has misunderstood the theory of the modal law-spheres in its fundamentals. Objects which have number have nothing to do with the modal structure of the numerical aspect. And numbers cannot be 'objects' in the sense of MEINONG's "Gegenstandstheorie" no more than apples, stones and other concrete things can belong to special modal aspects of meaning.
'Number' as such is a theoretical abstraction, a modal function, not a thing. The things in which numerical relations are inherent are not numbers, they have them. A set of things, viewed only according to the numerical aspect, is not itself a thing so that it can be an object of 'sensory perception'.
MALAN acknowledges that numbers are not individual things, but considers them as 'universal objects' or objects of the third stage (voorwerpen van die derde orde). Their species are not types of things, but only sets of things: They are to be distinguished from the genera whose species are determined by differentia specifica. A pair of shoes and a pair of twins are identical sets. Two sets are identical if each thing of the first set corresponds to a thing of the second. In other words, this identity is the one-one correspondence between the sets. This statement implies that, as far as their numbers as such are concerned, the things functioning in the sets are indifferent. It also means that in arithmetic the sets can only count for something as quantitative relations. Therefore the whole conception of 'pre-numeral sets' as 'species of universal numbers' is meaningless. RUSSELL conceived the one-one correspondence of the members of identical classes as a purely logical relation. But it is impossible to derive a quantitative equivalence from a purely analytical correspondence of members.
MALAN admits this. But his own view according to which numbers are genera of sensorily perceptible, pre-numeral sets of things is equally untenable. He overlooks the fact that a sensory multiplicity as such, abstracted from its intermodal relation to numeral multiplicity, is no longer quantitative in meaning. Consequently, numbers cannot be the genera of sensorily perceptible sets.
The modal meaning-nucleus of space. Dimensionality and spatial magnitude as arithmetical analogies in the modal meaning of space.
De modale zin-kern der ruimte. Dimensionaliteit en ruimtelijke grootte als getals-analogieën in den modalen zin der ruimte.
The spatial aspect in its original modality of meaning cannot exist without its substratum, viz. the numerical law-sphere. This will for the present be proved by means of a brief analysis of the modal structure of space in its original mathematical sense as regards its nucleus and its retrocipations.
Its original meaning-kernel can only be conceived as continuous extension in the simultaneity of all its parts within the spatial order of time. From the very beginning it must be clear that modern formal mathematics, in its theory of more-dimensional sets, has eliminated the spatial aspect as such. Spatial relations and figures are reduced here to special 'arguments' that play no essential role in the formalized theory. This has nothing to do with the discovery of the non-Euclidean geometries, but is only the result of the reduction of pure geometry to pure arithmetic, or to pure logic respectively.
From the philosophical point of view this elimination of the spatial aspect results in a premature elimination of the fundamental problem of the inner nature and meaning of pure space. This problem has been the subject of profound discussion since NEWTON, HUME, LEIBNIZ and KANT. But it has not found its definitive solution for lack of an exact analysis of the modal structures of meaning. The premature elimination of this fundamental problem has prevented the philosophy of mathematics from examining the primordial question concerning the original modal meaning of the spatial aspect of human experience.
In connection with this it is necessary to inquire into the relation between pure space and the analogical meanings of the spatial concepts used in all other sciences. It is the very task of the theory of the modal law-spheres to resume the study of this problem, which cannot be indifferent to mathematical theory.
We must especially warn against the identification of the original spatial meaning-nucleus with the objective sensory space of perception. The original meaning-kernel of the spatial aspect cannot be qualified by sensory qualities. Nevertheless, this modal nucleus cannot reveal its meaning apart from analogical moments which are qualified by it. In the creaturely realm of meaning even original kernels of modal aspects are bound to analogical moments in which they must express themselves. It will appear later on that even the meaning-kernel of the numerical aspect does not escape this universal coherence.
It is only as dimensional extension that we can grasp the original modal meaning of space. This original modal meaning is therefore dimensional continuous extension, so long as no account is taken of its anticipatory structure. Dimensionality, however, is an element of the spatial modality of meaning (viewed from its law-side ) which cannot exist without its coherence with the numerical aspect.
 Dimensionality, as such, does not imply a determinate magnitude of lines which, as the coordinates of a point, are constructed in different dimensions. It is only an order of spatial extension, not a determinate spatial figure. Therefore, it belongs to the law-side of the aspect, not to its subject-side.
As space may have two, three or more dimensions, it always refers to the arithmetical aspect as its substratum. Viewed from the modal subject-side of the spatial aspect, the spatial figure necessarily has its numerical analogy in its spatial magnitude. This retrocipation in the spatial meaning, so closely connected with the spatial point, will be analysed in our discussion of the modal subject-object-relation, because from this point of view it is highly interesting.
Provisionally it may be established that magnitude in the meaning of the space-aspect is only a retrocipatory analogy of number.
The so-called transfinite numbers and the antinomies of actual infinity.
Het zgn. transfiniete getal en de antinomieën der actueele oneindigheid.
Every attempt to transfer the moment of continuity in its original spatial sense into the modal aspect of number inevitably leads to antinomy. Such an attempt really implies the acceptance of the actual or completed infinity of a series, as was done by CANTOR, the founder of the theory of the so-called 'transfinite numbers' .
 G. CANTOR, Grundlagen einer allgemeinen Mannigfaltigkeitslehre, ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (1883).
This antinomy must come to light, if we accept transfinity in the orders of the infinite, and also if this actual infinity is assumed in the orders of the infinitesimal. The latter constitute a domain to which VERONESE has extended CANTOR's theory of the transfinite numbers in order to obtain a firm foundation for the whole of infinitesimal analysis. And the antinomy is implied in the fundamental concept of completed infinity itself, quite apart from the antinomic character of the different theorems that were supposed to be possible for the 'transfinite classes of numbers'.
The functions in the numerical aspect that anticipate the spatial,
kinematic and analytical modi.
De op ruimte en beweging anticipeerende functies
in den modalen zin van het getal.
In the infinite series, formed by the 'irrational' and differential functions of number, the modal meaning of the number-aspect undeniably reveals its anticipatory structure in that it approximates the original meaning of space and movement respectively. But it remains within the meaning-aspect of discrete quantity. The total of the discrete numerical values, functioning in these approximative series, can never be actually given in the anticipatory direction of time of the numerical aspect. In its anticipatory functions number can only approximate the continuity of space and the variability of motion, but it can never reach them. These meaning-functions of number are not to be considered as actual numbers. They are only complicated relations between natural integers according to the laws of number, just like the fractions and the so-called complex numbers.
In this sense I agree with the statement made by the intuitionist mathematician WEYL: "Mathematics is entirely dependent on the character of the natural numbers, even with respect to the logical forms in which it is developed" .
 "Die Mathematik ist ganz und gar, sogar den logischen Formen nach, in denen sie rich bewegt, abhängig vom Wesen der natürlichen Zahl." Cf. WEYL: Über die neue Grundlagenkrise in der Mathematik, in Mathem. Zeitschrift, 10 (1921) p. 70.
However, this does not entitle us to qualify the anticipatory, approximative functions of number as arbitrary products of the human mind, as is done by the intuitionist mathematician KRONECKER . They are rigorously founded in the modal meaning-structure of number and the inter-modal coherence of meaning.
 "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk." [Whole numbers have been made by God, all the others are the work of man], quoted by A. FRAENKEL, Einleitung in die Mengenlehre (2e Aufl. 1923) p. 172.
Only the interpretation of these meaning-functions as actual numbers is the work of man, but then work that mis-interprets the modal structure of meaning in the numerical law-sphere.
MALAN's defence of the concept 'continuous number'.
MALAN, in his treatise mentioned above, is of the opinion that discreteness and continuity are qualities which a number shows only in its relation to other numbers. The number 1 for instance can represent either a cardinal number, or a rational, or a real one. Whether a number is discrete or continous, depends on the question whether it is placed under the laws of discrete numbers or under those of continuous numerical values.
According to him this is only a question of the operator which is chosen. The choice of a particular selecting operator, as, e.g., + 1, is arbitrary. But the result of the operation performed with the aid of this operator is necessary, in conformity to the law of the function. The operator can only lay bare this law-conformity. Just as the discrete character of a number is laid bare by a particular operator of juxtaposition, so, according to the author, the continuous character is laid bare by an operator of repeated interposition or insertion.
I fear that MALAN has not grasped the point at issue. In the first place I must observe that not the operator itself, but only the choice of a particular operator, can be arbitrary. The operators + 1, + etc. are themselves implied in the quantitative aspect of time-order, and so is the operator of 'repeated interposition'.
When we choose the latter in order to find the series of 'real' numerical functions, it must be possible to indicate the law of the numerical series which is to result from the operation. If, however, this functional law implies that the process of interposition is necessarily infinite, then it implies at the same time that the quantitative series cannot be actually continuous. It will always be possible to insert new values between the members hitherto found. In other words, the fact that the process of insertion is continuous by virtue of the operator of 'repeated interpositon', does not guarantee the actual continuity of the series of numerical values resulting from the operation.
And the fact that the principle or law of the numerical series resulting from the irrational 'numbers' may be definite, does not imply that the latter have an actual existence as numbers on the same footing as natural integers.
MALAN cannot discover any anticipatory relation between the continuity of the process of interposing numerical values in the infinite series and the modal kernel of the spatial aspect: "It is inexplicable", he says, "how DOOYEWEERD can see something spatial in this continuous series." But I can explain why he cannot see it. This is due to the fact that he operates with an analogical space-concept without any critical analysis of the original nucleus of meaning of the spatial aspect as such. This is evident from the following argument which he directs against my analysis of this meaning-kernel:
"As regards space, there is of course continuity in space. But only an absolutizing metaphysician can declare that all kinds of space are continuous. As we have demonstrated in section I, there are, especially in the world of the sense of touch, discrete perception-spaces."
I never have said that 'all sorts of space' are continuous. In the analysis of the modal meaning-kernel of the spatial aspect we are not concerned with sensory space which can have only an analogical meaning, just like physical space, biological space, logical space, historical space and so on. But apparently MALAN conceives of the different modal 'kinds of space' as species of a genus. And this also shows that he has not understood the theory of the modal law-spheres. The latter is intended to lay bare the inter-modal relation between original kernels of modal meaning and merely analogical moments.
Number and continuity.
DEDEKIND's theory of the so-called irrational numbers.
Getal en continuiteit.
Dedekind's snedetheorie van het zgn. irrationeele getal.
The introduction of the element of continuity in the concept of number, — if not intended as an anticipatory, approximative moment of meaning, — is primarily to be considered as an effort to do away with the modal boundaries of the meaning-aspects of number, space, motion and logical analysis. Then the law of the continuity of the movement of thought, formulated by LEIBNIZ, is had recourse to for the purpose of rationalizing continuity in its original spatial meaning.
Such was the case in DEDEKIND's well-known attempt to rationalize the so-called 'irrational numbers', which prompted WEIERSTRASS, CANTOR, PASCH and VERONESE to make much more radical attempts in the same direction. The mathematician DEDEKIND would not look upon the continuity of the series as an anticipation of the meaning of space by the modal meaning of number. This would imply the recognition that the number-aspect is not self-sufficient in the anticipatory direction of time. By means of a sharp definition DEDEKIND wanted to introduce the idea of continuity into the concept of number itself as an original moment in the numerical meaning-aspect.
Now the 'irrational' function of number, which can never be counted off in finite values in accordance with the so-called Archimedean principle, was defined as a 'section' in the system of rational numbers.
How did DEDEKIND find this definition? At least in the first project of his theory he related all the values of the numbers of the system to points in a spatial line. Next he logicized these points in space into pure points of thought, which logical thinking subsequently again eliminates in the continuity of its movement. This procedure was based on the postulate that there is only one single definite numerical value corresponding to each 'section' of the rational system. The insertion of the 'section' fills a vacuum in the system, so that, if one imagines in thought that in this way all vacancies have been filled up, the whole system of numbers is without any gap, i.e. it is continuous. The modal boundary of meaning between spatial continuity and logical continuity seems to have been broken through in this method.
The complete theoretical elimination of the modal meaning of number, through the giving-up of finite numbers as the basis for the infinitesimal functions.
The modal shiftings of meaning in the logicistic view.
The modal shiftings of meaning in the logicistic view.
Algeheele theoretische uitwissching van den modalen getalszin in de prijsgave van het eindig getal als basis voor de infinitesimale functies. De modale zin-verschuivingen in de logicistische opvatting.
DEDEKIND at least took rational numbers and the Archimedean principle for his starting-point.
WEIERSTRASZ, CANTOR, PASCH and VERONESE, on the other hand, broke completely with the view that discrete quantity is the modal meaning of number. From the start they held the convergent infinite series, (in CANTOR: the fundamental series), to be an arithmetical concept. This they considered in its origin to be completely determined by arithmetical thought only and not bound to a deduction from the rational numbers by means of a 'theory of sections'.
PASCH introduced the very characteristic term 'Zahlstrecke' for the 'irrational number'. In this way he expressed that from the beginning the idea of original continuity has been included in the concept of number.
The Marburg school of neo-Kantianism has laid bare the inner relation between this whole rationalistic development of arithmetic and the creation-motive in the Humanistic science-ideal.
NATORP, one of the leading thinkers of this school, writes:
"In the last analysis it is nothing but the basic relation between the continuity of thought and the discretion of the separating act of thought which seeks and finds its definite, scientifically developable expression in the relation between number as a continuum and as a discrete quantity" .
 Die logischen Grundlagen der exakten Wissenschaften, p. 188: "Es ist zuletzt nichts als das logische Grundverhältnis der Denkkontinuität zur Diskretion der sondernden Setzung im Denken, was in dem Verhältnis der Zahl als Kontinuum zu den Zahldiskretionen seinen bestimmten wissenschaftlich entwickelbaren Ausdruck sucht und findet."
What strikes us especially in this statement is the exhaustive way in which this philosophical school logicizes the meaning-aspects of number and space. An elaborate system of shiftings has been applied to the meanings of these different spheres.
The original meanings of space and number are supposed to be deducible from the logical movement of thought in a process of logical creation. In other words, the original meaning-nuclei of number and space are first replaced by their analogies in the logical sphere: the arithmetical analogy of logical multiplicity, and the spatial analogy of logical continuity.
And, once this shift in the meanings of the aspects has been accomplished, it becomes possible to carry through the principle of the continuity of thought across all the modal boundaries of meaning. It stands to reason that in his way the meaning-nucleus of number can no longer be found in discrete quantity.
Then the point is how to find the logical origin of number in creative thought. This origin does not lie in the discrete finite one, but rather in the 'qualitative all-ness' (= totality) of the infinite .
 Die logischen Grundlagen der exakten Wissenschaften, p. 188.
The rationalistic concept of law in arithmetic.
Het rationalistisch wetsbegrip in de arithmetica.
This tendency in the Humanistic science-ideal to logicize the meaning-aspects of number and space made the rationalistic concept of law also subservient to its purpose. As a consequence the subject-side of the modal meaning of number was in theory completely merged into the law-side. Otherwise, it would never have occurred to anyone that the so-called irrational and the differential functions of the numeral aspect can be looked upon as real, actual numbers, and put on a level with the so-called 'natural number'.
Still less would the view have arisen that the discrete, finite numbers proper ought to be deduced from the infinite, if the subject-side of the law-sphere of number had not been theoretically merged into the law-side.
As observed above, an infinite series of numbers is no doubt perfectly determined by the law of arithmetical progression. This principle makes it possible a priori to determine the discrete arithmetical value in arithmetical time of any possible finite numerical relation in the series. For the rationalist conception of law this is a sufficient reason to attribute actual, completed infinitude to the series as a totality .
 NATORP, op. cit. p. 195/6.
But the identification of the law (in the definite principle of progression) with the actual subject-side of an endless series, is untenable. This is evident from the fact that in the infinitesimal functions of number the numerical modus in its anticipations approximates other meaning-aspects. But it is never able to exceed its modal boundaries in the anticipatory direction of time. After all, the numerical laws cannot be subjected to the basic arithmetical operations. But in arithmetic we must necessarily start from the natural numbers, if we are to work with irrational, imaginary, differential functions of number. The latter only deepen and open the meaning of the natural numeral values. The cosmic order takes revenge on the rationalistic trend of thought in mathematics which in theory eradicates the modal boundaries of meaning between number, space, movement (in its original mathematical sense) and logical analysis. As a result this thought gets entangled in the notorious antinomies of actual infinitude. [Aan de rationalistische, de modale zin-grenzen tusschen getal, ruimte, beweging en analyse theoretisch nivelleerende, richting in de mathesis wreekt zich de kosmische wetsorde door het denken in de beruchte antinomieën der actueele oneindigheid te verstrikken. (WdW Deel2 p70]
All these points ought to be more elaborately discussed in the special theory of the law-spheres. At this stage of our inquiry, we only wish to give a preliminary illustration of our method of analyzing the modal structures of meaning. The only intention is to shed light on the true nature and the coherence of the different elements of meaning in contrast with the prevailing rationalistic currents in mathematics.
Herman Dooyeweerd, New Critique of Theoretical Thought, Vol II/ Part I/ Chapt 2/§3 pp 83-93)
Às an leabhar "An Introduction to Christian Philosophy" le J.M. Spier