Patterns of Conceptual Integration

Number and logic

Anthony Judge

7.1 Beyond 2-term logic: multi-term systems: In the above argument the terms "set" and "system" have been used interchangeably since one of the characteristics of the sets of elements under consideration was identified as the complementarily of their elements. In discussing multi-term systems, a mathematician and director of industrial research J. G. Bennett clarifies further the kinds of sets to which these arguments apply (45, vol. 2, pp. 3-10). A set of elements taken without reference to any internal connections is called a class. A system is to be distinguished from a class, and he suggests rules for doing so [34, 35]. His summary of the characteristics of systems clarifies the definition of the sets considered here (see Annex 1).

Bennett notes that: "The properties of systems are usually studied in terms of their inner connectedness, but there is no general doctrine of systems based upon the properties that are associated with the number of terms by which they are constituted. This is strange, for philosophers have always been deeply concerned with the question whether or not there is a fundamental number system in the basic structure of reality." (45, vol. 2, p. 4) Such systems have however always had to be studied by using the conventional two-term logic. "Our usual language, though full of inconsistencies and ambiguities, can be adapted to the description of two-term systems. When the meanings of words and sentences are defined with special care, a logic is constructed that turns out to be the law of two-term systems. . .

The ambiguities and inconsistencies of our ordinary speech are not a defect, and recognition of them is a reminder that experience has more dimensions than logic. Analytical and sceptical philosophers have, during a hundred generations, exposed the barrenness of two-term thinking, and it becomes necessary to examine the possibilities latent in higher modes of thought." (45, vol. 1, p. 25).

In support of this investigation Bennett quotes Bertrand Russell on classical two-term logic: "The extension of the subject-predicate logic is right as far as it goes, but obviously a further extension can be proved necessary by exactly similar arguments. How far it is necessary to go up the series of three-term, four-term, five-term relations, I do not know. But it is certainly necessary to go beyond two-term relations." (46) Bennett draws attention to the widespread dualism in thought, feelings and instinctive reaction [36] and comments, as an example, on the difficulty of triadic (three-term) thinking. "Contemplation of the triad is not merely recognizing a third idea as the reconciliation of two contradictories, but rather seeing in the union of the three an exemplification of the fundamental relationship by which all experience is governed. So long as nothing more is at work than the primitive associating mechanism, to speak of the 'unity of the triad' conveys little meaning. In order to perceive this unity directly, a power of attention is required that comes only with a change of consciousness." (45, vol. 1, p. 26) He notes Russell's view that it appears to be beyond the ordinary power of man (47), although clearly both Bennett, and others believe that there is a way around the limitation (42, 48, 49).

As an indication of the route to be followed, Bennett remarks that:

"The doctrine of logical types indicates that some words do not refer to terms but to systems. For example, a single term may have qualities, but it cannot have relationships. Relationship is the property of a system, and at first sight it might seem that any multi-term system can exemplify relationships. It can readily be seen that a dyad--that is a two-term system-- cannot carry a relationship . . . If relatedness is a property or quality that belongs to three-term systems, the question arises whether there are other properties that characterise systems with different numbers of terms" (45, vol. 2, pp. 5-6).

The point to be emphasized as a result of the above argument is that the sets fundamental to the social sciences and policy-formulation constitute systems whose characteristics merit investigation irrespective of the nature of the terms in any particular case. Namely a 5-term set of values (concepts, principles, problems, etc.) has characteristics distinct from a 7-term set, irrespective of the values selected in either case. And, furthermore, such characteristics are solely dependent upon the total number of terms in the set.

7.2 Logic of inter-paradigmatic dialogue: In proposing a deliberately non-western complement to the Aristotelian logic of western science, Kinhide Mushakoji (49) introduces a third pole in the dialogical process to destabilize the intellectual equilibrium which exists between two paradigms dividing a given intellectual community into two opposing poles. He then argues, in the light of complementarily in physics, that:

"inter-paradigmatic dialogues not only in natural sciences but also in social sciences - should be concerned not with the determination of who is right or wrong in defining a concept one way or another. It should rather concern itself with the question of what parts of the natural or social realities are best approached by one or the other position. Two formally contradictory definitions of the realities may be both relevant and complementary in shedding light on different aspects of the same social realities. This is why the logic of inter-paradigmatic dialogue cannot be bound by the laws of Aristotelian formal logic: identity, contradiction and excluded middle....

A group theoretical treatment of concepts used by a given paradigm is insufficient because it deals only with the structure of the significant system (the logical level) without touching how the signifié realities (the reality level) are decomposed when one relies on a given paradigm (50).

This "logico-real" problem of the relationship between the logical and the reality level calls for a study of the morphogenesis of paradigms. Catastrophe theory helps us here since it sheds light on the different logical positions in the morphogenetical space. A major difference between the two levels of "significant" and "signifié" lies in the fact that the former is composed by discrete concepts while the latter is a continuous space. Therefore, it becomes necessary to apply a catastrophe theoretical model relating the continuous reality (i.e. the "signifié") with the discrete set of concepts (i.e. the "signifiant"). "

His reference to catastrophe theory, formulated by Rene Thom (32), relates this argument to that on logical "curvature" (above). Mushakoji then argues for a nonformal logical model developed in oriental logic on the basis of four lemmas (affirmation; negation; non-affirmation and non-negation; affirmation and negation). Such lemmas are concerned with the modalities according to which the human mind grasps reality rather than how human intellect reasons about it (51). He considers the lemmic approach to be a breakthrough in view of the possibilities it provides for overcoming the static ontology of the West inherited from Parmenides and highlighting the limitation of means-end rationality.

Mushakoji's concerns are shared in part by Sallantin (48) and Varela (see above), although they both elaborate on 3-term systems in much greater detail. The relationships between these three is elusive and a broader framework touch as Bennett's) raises questions: (a) of the possibility of 4, 5 or higher term systems, (b) of why the three authors are seemingly insensitive to the qualitative attributes of systems higher in the series and (c) of the implication for classification.

7.3 Number and N-term systems: In order to make further use of the programmes that Bennett and von Franz respectively set themselves, it is necessary to link the concept of N-term system (Bennett) and that of number as studied by von Franz. What these and other authors have each attempted, in one way or another, is to identify the qualitative characteristics to be associated with each term in the series:

  • one-ness, two-ness, three-ness, etc. or unity, duality, triplicity, etc.
  • or one-term, two-term, three-term, etc.
  • or monad, dyed, triad, etc.
  • or unitary, binary, ternary, etc.

Bennett argues the case as follows:

"Even when enfranchised from the limitations of logic, thought does not reach beyond the triad; yet we cannot doubt that four-term, five-term, and even higher systems must be significant ... Multi-term systems oblige us, therefore, to take account of the significance of number as a factor in all experience; and for this we must seek a fuller apprehension than is given by logic. The logical interpretation of number derives from the formation of classes, and is essentially polar or dualistic; that is, it consists in the assignment of an object to a given class in terms of the simple distinction of 'yes, it is a member' or 'no, it is not'. This procedure leads to a view of number according to which there is nothing to be known except the laws of arithmetic. These laws belong, however, merely to a primitive form of logical thought." (45, vol. 1, pp. 26 - 7).

Sallantin also addressed this point (48).

Bennett argues that there are several other ways in which we can think about number, such as lead to cardinal or ordinal numbers. In addition the 'arithmetic quality', based on the inner relationships of a group, may be used to distinguish prime and composite numbers, for example. But even so

"the full significance of number is far from being exhausted. Numbers have meaning in their own right. The number two is not merely the symbol of duality; 'twoness' depends upon and defines the separation of opposites. The number three is indissolubly connected with the very idea of relatedness. Three as a class concept is an abstraction from experience - three as a relationship is an integral part of experience itself. This leads us to seek for a property which can be called the concrete significance of number." (45, vol. 1, p. 28)

Bennett joins von Franz in recognizing that: "The search for the concrete significance of number is very ancient... At some unknown period ... man had already become convinced of this concrete significance, and must, therefore, have seen how a number can enter directly into events as experienced by himself." (p. 28) And: "If we are ever to free ourselves from the limitations of logical thinking, we shall have to discover a new significance in number; for number and logic, as we know them today, are inseparable." (p. 28)