mardi, décembre 28, 2010

Dooyeweerd: ÀIREAMH (NUMBER)

"Boireannach le Biorsamaid" le Johannes VERMEER (mu 1665)
§ 3 - Voorloopige analyse van de drie eerste modale zin-structuren.
A - Brief analysis 
of the original meaning of 
A - Summiere analyse 
van den originairen zin van het 
    Although the systematic analysis of the modal structures of meaning can only be treated in the special theory of the law-spheres, we will now put our conception to the test by the analysis of some of them.

The original nuclear meaning of number, 
and the numerical analogy in the logical modality of meaning.
De originaire zin-kern van het getal 
en de getalsanalogie in de logische zin-modaliteit.

     When we try to analyse the modal meaning of the numerical aspect, it is necessary to start with the natural cardinal numbers, in which this meaning discloses itself in its primitive and irreducible structure. For all the rational, irrational and complex numeral functions in the last analysis pre-suppose the natural numbers [1]
[1] Even MAX BLACK (The Nature of Mathematics, p. 38) states in his examination of the formalizing of pure mathematics in logistic: 'Arithmetic is in a peculiar position, since definite integers occur in all systems of axioms, but even that subject can be arranged as above to begin with axioms whose subject-matter consists of integers and relations between integers.' And a little further on he says: 'This apology for formal analysis requires two important reservations in the case of pure mathematics. (1) The natural numbers as we have just seen are in the peculiar position of occurring as constants in all axiom systems, and therefore marks denoting integers must be understood in a sense in which lines, points, etc. need not be understood. (2) No complete axiom system can be set up for 'real numbers'. That is to say in the two cases where the fundamental philosophical analysis of mathematics arises it will be found that no 'formal' analysis is adequate.' (p. 39/40).
Every attempt to reduce the modal meaning of the latter to purely logical relations rests, as will appear, on a confusion between numerical analogies in the structure of the analytical relations and the original kernel of numerical meaning. The latter can be found in nothing but quantity (how much) disclosing itself in the series-principle of the numerical time-order with its + and — directions. This modal time-order itself is determined by the quantitative meaning of this aspect. KANT denatured the nuclear moment of the numerical aspect to a transcendental logical category, though he derived the different numbers from the so-called schematizing of this category in time (as a transcendental form of sensory perception).
     The view, however, that arithmetic is no more than a special branch of logic, has indeed been prevalent since the Humanistic science-ideal developed the idea of the "mathesis universalis". Many students of logistic suppose they possess in this splendid instrument of human thought all the requirements to deduce the number concept in a purely analytical way from the general logic of relations.
     Now the logical modality of meaning has for its irreducible nucleus the analytical manner of distinction (or distinctiveness, respectively, when the analytical relations are viewed as modal subject-object-relations referring to the analytical characteristics of things). In the structure of this modality there is indeed an analogy of number to be found. This analogy, however, receives its determinateness of meaning only in the nucleus of logical meaning itself. This numerical analogy is the analytical unity and multiplicity, inherent in every analytical relation and in every concept according to its logical aspect. Every concept, viewed logically, is a σύνθεσις νοημάτων, the logical unification of various logical moments into an identical unity. The unifying-process develops according to the analytical norms of thought, viz. those of identity and contradiction.
     Every analytical relation, even that of identity, implies a numerical analogy, because analysis itself is a manner of distinction, and distinction implies at least two terms: the one and the other.
     As a numerical analogy the logical unity and multiplicity remain qualified by the analytical nucleus of logical meaning. But they undeniably refer back to the original nuclear meaning of number proper in the coherence of meaning of cosmic time.

The relation between number and logical multiplicity.
Verhouding van getal en logische menigvuldigheid.
     Logical unity and multiplicity, just as logical allness, are necessarily founded in the meaning of number, and not vice versa [2].[De logische eenheid en menigvuldigheid is noodwendig in den getalszin gefundeerd enniet omgekeerd.]
[2] This is involuntarily admitted by A. RIEHL (though he takes the view, in accordance with KANT, that an abstract number is an a priori scheme of the logical category of quantity), when he says (Der Phil. Kritizismus, 2e Aufl., 1925, II, S. 15): "Durch alle Verschiedenheiten der Vorstellungen hindurch, über alle Unterbrechungen des empirischen Selbstbewusztseins hinweg erhält sich das eine: Ich denke, als numerisch mit sich identisch." [Through all varieties of representations, over all the interruptions of the empirical self-consciousness one thing remains intact: 'I think', as being numerically identical with itself]. From this it follows, that even KANT's concept of the transcendental-logical unity of apperception, assumed to be the foundation of the 'category of quantity', appears not to be detached from the meaning of number. On the other hand, number is called "eine Schöpfung unseres Geistes" [a creature of the mind]! (ibid., p. 96).
     The logical characteristics, summarized in the unity of the concept, cannot be a logical multiplicity if they do not have their number [aantal]. The fact that this cosmic order of time between number and logical multiplicity was lost sight of, can be explained in some writers because they deduced number from the subjective human act of counting. Counting is naturally impossible without analytical distinction. But is number in its original sense only the product of counting? This supposition cannot be correct, since every act of counting pre-supposes an at least implicit pre-theoretic sense of the meaning of number and its inner conformity to law.
     Moreover, logical multiplicity is qualified in a modally analytical way. This multiplicity, in any case, is a dependent moment in the modal structure of the analytical aspect, deriving its qualification from the analytical nucleus of meaning.
     A modal meaning-moment, lacking the qualifying character of a nucleus, can never be original, but always refers to another meaning-nucleus lying outside the modal aspect concerned. Logical multiplicity is a retrocipation to a substratum, and not an anticipation. This appears from the fact that the analytical meaning-nucleus always pre-supposes a numerical multiplicity, even in pre-theoretical thought. This is why numerical quantity must find its analogy in a modally logical sense in analytical multiplicity. In the pre-theoretic, naive understanding the first multiplicity to which analytical distinction appeals, is of an objective sensory-psychic nature. Pre-theoretical distinction rests upon a primitive analysis of a perceived sensory multiplicity. But also this sensory multiplicity cannot be the original manifold. It must refer to an original multiplicity in the sense of discrete quantity. Animals cannot arrive at a logical concept of number. But they certainly have a sensory perception of multiplicity, which latter can in no case be of an analytical character.
     And finally, the method of antinomy can be applied to the attempt to ascribe the original meaning of number to merely logical multiplicity.
     The proposition 2 + 2 = 4 is true in the (theoretically grasped) original numerical meaning. But we should not try to deduce this addition only from analytical thought after the manner of logistic with the aid of the concept of class [3].
[3] Cf. e.g. B. RUSSELL. The Principles of Mathematics, Vol. I (1903) p.119: "The chief point to be observed is, that logical addition of cIasses is the fundamental notion, while the arithmetical addititon of numbers is wholly subsequent." The deduction of number from the class-concept was first attempted by FREGE.
For then it appears that we get entangled in patent antinomies due to the theoretical attempt at erasing the modal boundaries between analytical and numerical multiplicity. Besides, there arises a vicious circle with respect to the cosmic temporal order of the two modal aspects concerned. The reason is that the extension of a class-concept presupposes number in its original sense [4].
[4] That is to say the reduction of the integers to the analytical class-concept is not merely a tautology, which has a quite legitimate function in formal analysis. But it rests upon a fallacious ὕστερον πρότερον with regard to the cosmonomic place of the numerical and the analytical aspects.
     The antinomy, implied in the attempt here intended, can be demonstrated as follows. The sign + is indeed the linguistic symbol signifying the positive direction of the temporal order in the originally quantitative sense of number. In the successive progress of counting the new addition of numbers in the + direction supposes a greater positional value in the series. The two first integers after 0 are really earlier in a quantitative sense than the two next added to them, because their positional value is smaller. The third added unit has the positional value 3, the fourth the positional value 4. If, however, it were allowed to interpret the + sign in an original analytic sense and not in an original quantitative meaning, the judgment 2 + 2 = 4 would per se be in conflict with the principium contradictionis. For, whichever way we turn, from a merely logical synthesis of two numbers there can never arise a new number. KANT saw this very clearly [5].
[5] Kr. der reinen Vernunft. Einleitung S. 45 (W.W. vol. V, Grossherz. W. E. Ausg.).
     If logistic tries to avoid this antinomy by executing the operation of a 'logical addition' on classes and not on the numbers themselves, it moves in the vicious circle mentioned above. Let us consider the latter more in detail.

Number and the class-concept. RUSSELL.
Getal en klassebegrip. Russell.
     RUSSELL, with WHITEHEAD one of the best philosophically trained mathematicians of this movement — admits that the logical addition of 1 and 1, according to the principles of symbolic logic, would always yield one as its result. That's why he gives the following definition: "1 + 1 is the number of a class -w- which is the logical sum of two classes -u- and -v- which have no common term and have each only one term" [6].
[6] Principles, p. 119.
     But it may be clear already in the present context that the antinomy RUSSELL tries to avoid by introducing the class-concept, reappears in the vicious circle of his definition.
     RUSSELL tries to deduce the concept of number from the extension of the concept of class. But for the simple distinction of the classes he needs number in its original meaning of quantity [7].
[7] This is also argued by CASSIRER, Substanzbegriff und Funktionsbegriff (1923) p. 66, who rightly rejects RUSSELL's defence against this objection.
In other words, RUSSELL's definition of the sum 1 + 1 remains burdened with the inner antinomy whose existence he himself admitted in the attempt to deduce the number 2 from a 'logical addition' of 1 and 1.

Herman Dooyeweerd, New Critique of Theoretical Thought, Vol II/ Part I/ Chapt 2/§3 pp 79-83)