Number and Logos
Unforgettable Hours with Warren St. McCulloch
(Part 2 of 4)
|It is the purpose of this
essay to present the author's theories but to show the philosophic profundity of McCulloch
and the author's spiritual indebtedness to him. So we shall return to the remarks
McCulloch made about subterranean relations between arithmetic and the hermeneutics of the
humanities. From Dilthey McCulloch went back to Hegel as idealist and materialist were
equally untenable because Idealism and Materialism both implied that they were sets of
statements about what there is instead of what the universe means to the
brain. In any case Hegel's philosophy recognizes an existence as a context of stateable
facts. In this respect Hegel was still dependent on Immanuel Kant who "spawned two
fertile succubi"' as we read in "The Past of a Delusion", One was "the
Dynamic Ego as Unconscious Mind. Upon (it) Freud begat his bastard, Psychoanalysis. The
other, causality, the Category of Reason, flitted transcendentally through Hegel's
Dialectical Idealism." Upon Causality herself Karl Marx begat his bastard,
Dialectical Materialism. "The author being a stout defender of the Theory of
Dialectics then asked McCulloch whose opinion of dialectics in the "Embodiments of
Mind" seemed to be extremely low whether dialectics would play a role in a not
ontological, but hermeneutical alternative of idealism and materialism. McCulloch conceded
that there might be something to it provided a satisfactory interpretation could be
found for the "indeterminate duality"
of Greek philosophy. According to
Aristotle's metaphysics Plato called the forms numbers and stated that each number has two
constituents: the One or unit which Aristotle defines as the formal constituent; and
something which he calls a material constituent. This is supposed to be the mysterious
. It stands
to reason, of course, that dialectics has its root in a duality. So a renewed and critical
analysis of dialectics should start from here. McCulloch seemed to be very well versed in
these antecedents of number theory but he voiced some doubt whether the problem of the
indeterminate duality was as yet properly understood. He was ready to admit that the
testimony of Aristotle seemed to be unimpeachable with regard to what Plato said but
it seemed to be a different question as to what Plato really meant. The author who
had studied the relevant passages in Aristotle's metaphysics could not help imparting to
McCulloch his impression that Aristotle totally misunderstood Plato's reflections
concerning the theory of numbers. Aristotle himself refers to the lectures Plato delivered
in the Academy as the
"unwritten doctrine" which means that Plato did not produce a written text of his academic teaching. Therefore his listeners handed on several different versions of his famous lecture on "the Good" which has intrigued students of Plato up to the present time.
McCulloch was intimately familiar with Alfred North Whitehead's essay "Mathematics and the Good". Whitehead keeps quite close to the tradition which connects the Platonic "duality" with the "indefinite" or the "unlimited" of the Pythagoreans. Whitehead interprets this in the following way:
"The notion of complete self-sufficiency of any item of finite knowledge is the fundamental error of dogmatism. Every such item derives its truth, and its very meaning, from its unanalyzed relevance to the background which is the unbounded Universe. Not even the simplest notion of arithmetic escapes this inescapable condition for existence." ("Essays in Science and Philosophy" 1947, p. 101.) McCulloch could not agree entirely with this viewpoint. Seymour Papert correctly pointed out that the famous 1943 paper by McCulloch and Pitts demonstrated that a logical calculus that would permit the embodiment of any theory of mind had to satisfy "some very general principle of finitude". McCulloch was thinking of some such limitation in the indeterminateness of "indeterminate duality" when he questioned the traditional and conventional interpretations of Plato's ideas on numbers. It was clear to him that in this respect the difference between Plato and Aristotle is basically this that Aristotle permitted only one single concept of number, producing a gradual accumulation of uniform units , but that Plato's philosophy involved a second concept of number resu1ting from the break between the real of ideas and our empirical existence. He became very insistent that the author should delve deeper into the philosophical aspects of number theory when the latter told him about Hegel's speculation on a "second" system of mathematics "welche dasjenige aus Begriffen (erkennt), was die gewöhnliche mathematische Wissenschaft aus vorausgesetzten Bestimmungen nach der Methode des Verstandes ableitet". (Hegel, ed. Glockner IX, p. 84.) With this idea of a "second" system of mathematics in the background McCulloch began to urge the author to develop his ideas on the connection between number and logical concept further. Very soon an agreement was reached that the starting point should be the fact that the notation of the binary system of numbers coincided in an interesting way with the method by which two-valued truth tables demonstrated in the propositional calculus the meaning of logical concepts like conjunction, disjunction, implication and so on. It was only necessary to reduce the value sequences to their underlying morphogrammatic structures of which eight could be obtained in order to see that there was a peculiar correspondence between the method by which the binary numbers from 0 to III were produced and eight 4-place morphograms which used only the idea of sameness between places or difference.
We do not have to repeat all of the next steps here because they have, almost without philosophic background, been reported by the author in Vol. I. in the Journal of Cybernetics. Almost - which means that the formal philosophical concept of universal contexture at least was introduced. But neither Plato's nor Hegel's idea of a "philosophische Mathematik", as logically distinct from traditional mathematics, was alluded to. There was also no reference to a general principle of finitude which had been most essential for the production of the afore-mentioned essay in the Journal of Cybernetics. In fact, the essay could never have been written without the information the author was given by McCulloch about some of his ideas on finitude. The author shall try to repeat what his memory retained because what McCulloch developed in the case of the dialogue seems to deviate from the trend of thought emerging in the "Embodiments of Mind".
After a tentative discussion of Hegel's trans-classic concept of mathematics McCulloch turned back to the problem of finitude referring to a then recent paper by C. C. Chang "Infinite-valued Logic as a Basis of Set Theory". (Logic, Methology and Philosophy of Science, North Holland Publishing Company, Amsterdam, pp. 93-100, 1965.) He agreed with the author that Chang's paper had to be criticized from the viewpoint of finitude, and that Chang assumed willy nilly the philosophical theorem of Lukasiewiez that only three systems of logic have ontological relevance: the two-valued system, the three-valued order and a system with an infinite number of values. He admitted that Lukasiewiez's conclusion was quite consistent and reasonable provided one places all values added to True and False "between" these two classical boundary cases of value. That a two-valued logic and a system with an infinite number of values have ontological relevance is beyond question. But why in addition to them only a three-valued system? This assertion of Lukasiewicz may be interpreted as follows: Since the number of values between True and False represents the continuum, any individual value in the middle that is selected out of the totality of values can only be obtained by a Dedekind cut. This cut, and not the number obtained by it, is the proposed third value! Thus, if we add a fourth and a fifth and a sixth and so on intermediate value we would only iterate in logical respect the information of the cut. And since - to say it again - the cut itself is the third value and not the results of the cut. The iteration of the cut would, despite a different numerical result, produce logically (and not arithmetically) speaking the same value. Seen from here it makes sense, if Lukasiewicz maintains that only to three systems of logic philosophical meaning can be attached. The talk then turned to the fact that the author had shown in several papers that many-valuedness might be interpreted differently. Denoting all values by integers and starting with 1 one might place all transelassical values not "between" 1 and 2 but 2 "beyond" 2. This "beyond" leads inevitably to a different interpretation of many-valued systems.
At this point the author wants to note that during the initial stage of investigating many-valuedness he had believed that the idea of placing additional values totally beyond the alternative of True and False was the only legitimate ontological interpretation of many-valuedness. It was McCulloch who disabused him of this erroneous belief. He drew his attention to the fact that in a many-valued system designed according to the author's concept of many-valuedness being an order of ontological places of two-valuedness any two-valued system could additionally contain Lukasiewicz' values between True and False. Later on the author has found this suggestion extremely useful and only recently it has helped him to understand a specific phenomenon of trans-classic logic which, otherwise, might have been uninterpretable.
At this time, however, the new insight in many-valuedness did not lead very far. For the time being there existed only a general agreement between McCulloch and him that the term 'many-valuedness' was ambiguous. The theory had to consider the fact that two different kinds of many-valuedness had to be distinguished (1). Beyond this result there was still much haziness. It was about the time when McCulloch was playing with the idea of the "Triads" (2) , and the author distinctly remembers the day when McCulloch told him: "Gotthard, you can do everything with triads!" The author did not agree; there was too much of the small of Post and Lukasiewicz around this statement. However, he remained silent; McCulloch sounded too emphatic. It must have been the right diplomacy, because later - the author cannot remember the length of the interval - McCulloch declared with equal emphasis when the author based an argument on three-valued relations: "Triads are not enough". The author can guess what caused this change of attitude. First, the return of the discussion to the paper of Chang, and second, a renewed analysis of the meaning of number in the Platonic system. We shall start with Chang. He introduces in his paper a set X which is referred to as the set of truth values of the infinite-valued logic. For the purpose of discussing finite-valued logics he considers a sequence of finite subsets of X, such that for each Xn
Each set Xn, is recarded as the set of truth values of an n-valued logic. If n=2, all functions will, of course, acquire their traditional two-valued character and meaning. The viewpoint underlying this procedure is exactly the same as taken by Lukasiewicz. All values of this pseudo-transclassic logic have their ontological location between the boundary values 0 and 1. In other words: they refer to finite subsets of the continuum. This makes it impossible to eliminate infinity from the basic philosophic theory of logical values.
On the other hand, human awareness as the source of logical-value-and-natural-number theory is a finite system of the brain ("Why the mind is in the Head"). Although the system is finite it may produce as its mental content such second order concepts as denumerable and non-denumerable Infinity. If the author understood McCulloch properly then the latter took an extremely revolutionary position. Hitherto philosophers had always - without further questioning - assumed that the Finite is embedded in what we call the Infinite McCulloch seemed to imply that this order should be reversed and that infinity should be robbed of its primordial rank and only be admitted as a second order product of a finite system of awareness which is a product of the equally finite system of the physical brain. It became clearer and clearer to him that McCulloch's ultimate concept of the entities which made up Reality was not so much the Realm of Ideas - be that in the Platonic or in the Aristotelian-Hegelian sense - but the 'Pythagorean" conception of Number although his notion of numerosity had, in the course of the years, drifted away from the position which was taken in "What is a Number, that Man may know it". So at least it seemed to the author. When he first meditated about number it happened against the as yet unquestioned metaphysical backgrond that in order to define Reality one must understand that all Finitude is embedded in the Infinite. When the author saw him last McCulloch seemed to have completely reversed his position. He seemed to believe that ultimate Reality could only be understood in terms of Finitude, and that Reality conceived as infinity was nothing but mythology. The author was led to this conclusion by the discussion of Whitehead's "Mathematics and the Good". Which, of course, led directly to Plato's lecture and the modern attempts to reconstruct the text.
Plato starts with the question: what are the ultimate building stones of the Universe? The conventional interpretation of Plato is satisfied with the somewhat crude answer that these building stones are the Ideas. But if the ideas represent no ordered system in the shape of a pyramid, with the single idea of the Good on top, and a plurality of other ideas below, the problem of the metaphysical Number emerges and we are carried beyond the domain of Ideas to the ultra-ultimate question: what is the relation between unity and the manifold? In other words: our thinking cannot stop till it reaches the concept of what is conventionally and vaguely known as the natural number. It was immediately clear to McCulloch that our conventional interpretation of the order of natural numbers as a Peano sequence could not satisfy the philosophical reflexion because it was abstird to interpret the order ot the Ideas also as a Peano sequence. From the idea of the Good they spread out in an arrangement that was more or less inadequately described as a pyramid. The reports on Plato's lecture unfortunately do not make it clear how Plato himself interpreted the relation between Number and Idea. McCulloch as the cyberneticist interpreted it for purely systematic reasons as a reduction. The analysis of the Ideas leads to a pre-ideative system of only numerically definable relations. An alternative interpretation - traceable back to antiquity - that Ideas are just numbers he did not like. The ideas could not be the ultimate building stones of the universe - they were much too complex. It was unfortunate that neither McCulloch nor the author were aware of the fact that shortly before they entered into their discussion about natural numbers the German philosopher Klaus Oehler had published (in 1965) a paper under the title "Der entmythologisierte Platon" Zeitschr. f. Philos. Forschung XIX, pp. 393-420). This profound essay seems to have anticipated McCulloch's position. What Oehler says is so important that it may be repeated at this point. "Die Entfaltung der Einheit zur Vielheit und die Teilhabe des Vielen an dem übergeordneten Einen bestimmen den gegliederten Aufbau des Ideenkosmos. Nun geht aber weder der Aufstieg zu den umfassenden Begriffen ins Unendliche fort, noch geschieht das bei dem Abstieg zu dem Einzelnen. Der Aufstieg ist begrenzt durch den allgemeinsten und umfassendsten Begriff, das , der Abstieg ist begrenzt durch das jeweils letzte . Das bedeutet aber, daß die Ordnung der Ideen zahlenmäßig bestimmt ist. Folglich ist jede Idee durch die Zahl von Inhalten, die sie umschließt und an denen sie teil hat, eindeutig festgelegt. Jede Idee ist also durch eine Zahl bestimmt und ist als solche zahlenmäßig bestimmbar, angebbar. Diese numerische Fixiertheit verleiht der Ordnung der Ideen ihre rationale Klarheit, ihre Durchsichtigkeit und Übersichtlichkeit. Ist das Mannigfache der sinnlichen Wahrnehmung nur durch die Teilhabe an der Idee das, was es ist, so ist die Idee nur dureh die Teilhabe an der Zahl das, was sie ist. Mithin muß die Zahl vor der Idee sein. Die Ordnung der Zahlen ist der Ordnung der Ideen übergeordnet, weil überlegen. Das bedeutet aber: die Ideen sind nicht das Letzte und mithin nicht die Prinzipien des Seienden."
(The unfolding of the one into the manifold and the participation of the manifold in the super-ordinated One determine the structure of the cosmos of Ideas. But neither does the ascent to the comprehensive concepts continue into infinity, nor does this happen in descending to the Particular. The ascent it limited by the most general and the most comprehensive concept, the , the descent is limited by the last particular . That means that the order of ideas is numerically determined. If follows that each idea is univocally defined by the number elements it contains and in which it participates. Consequently each idea is characterized by a number and is as such numerically describable (and quotable). This numerical fixation endows the order of ideas with its rational clarity, transparency and orientability. If the manifold of sensual perception is what it is only by participation in the idea, then the idea is what it is only by participation in Number. Thus Number must be prior to Idea. The order of Numbers is superordinated to the order of Ideas, because it is more potent. This means: the ideas are not ultimate and therefore not the principles of Being.)
It is not difficult to see that Oehler leans toward the notion of finitude, which was so dear to McCulloch, when he points out that the ascent to the One as well as the descent to the Particular are always finite. That does not exclude, of course, that each such finitude may be superseded by numerical increase of the finitude. Infinity, however, is nothing but the everlasting subjective expectation that every given finitude is not the last one. It is a mistake to ascribe ultimate ontological relevance to the concept the Infinite. It seems to the author in retrospect that Mculloch in expressing such thoughts moved into the neighborhood of mathematical intuitionism and its criticism of the transfinite or actual (extensional) non-finitude. Existence is constructibility, logically speaking.
Before we discuss the quotation from Oehler it will be not only desirable but necessary to introject into the report on McCulloch an excurse on the meaning of the term 'number'; because a modern mathematician will probably object to the way this concept has been handled so far not only by McCulloch but by the author and Oehler as well. The question one has to begin with is the following: why did the concept of number become so important for Plato after the doctrine of Ideas had reached some maturity? The likely answer is, that during the development of the doctrine of Ideas, the quest for the individual ideas lost more and more of its importance in favor of the inquiry into the inter-connectivity and systematic order of all the ideas. This led automatically to the search for the most general and, at the same time, elementary form of order. This would, of course, be the linear order mentally accomplished by the simple process of counting. But already the Pythagoreans had discovered - and Plato was familiar with Pythagorean number theory - that this most primitive order was capable of a highly sophisticated treatment which permitted ultimately to encompass any element of ordering the not-yet-ordered.
Such concept of order transcends the principle of quantity by far and such transcendence may be determined in many ways. McCulloch only insisted that any principle of order should be traceable back to the familiar order of natural numbers. Whether we let the natural numbers begin with 0 or 1 is, of course, a mere convention. However, there should be no confusion between the metaphysical Nought and the conventional 0 or 1 in numbers. These distinctions remained in the discussions with McCulloch always somewhat vague; but he left no doubt that he never considered the gap between number and concept as ultimate but was convinced that it could be bridged. This was for him the significance of transcendental philosophy which he believed would produce the unification of the humanities and the sciences. Both of them - so he argued - start from a common ground: the elementary unit which in its primordiality is indistinguishable from any other unit. Thus primordial units are per se unordered and for this very reason they may be used to produce a system of order for the Realm of Ideas. But even at its very beginning Greek mathematics encountered an almost unsurmountable problem: how to understand the relation between unit in the geometrical and in the arithmetical sense. In the Pythagorean mathematics of the fifth century the geometrical point was made to correspond to the arithmetical meaning of I. In other words: the number I that which designated a real point in the objective world. A point is the minimum quantity which we encounter. The difficulties that arose from this viewpoint are too well known to mention them here; it is sufficient to draw the attention to the fact that Aristotle nailed this epistemological attitude down with the formula (the unit with location).
At this point the dialectical mechanism of all reflection makes itself visible, and the argument emerges that a point as identified with the number 1 is not a minimum volume of objectivity, but the absence of objectivity. In other words: to produce as number as a quantity a duality is required. As soon as this insight is obtained the thought will tend to let the point correspond rather to 0 and not to 1.
If in modern times we insist that it is irrelevant whether we call the first number 0 or I, this may be a convention in one way; but it is not a convention in a different way because it points to the peculiar relation between primordial unit and Nought.
It would be tempting to spin a consistent yarn how McCulloch connected his many philosophical ideas on Number with each other. Yet this would falsify the situation and the author refrains from doing so.
(1) Cf. G. Günther, Die
Theorie der ,,mehrwertigen" Logik: in Philosophische Perspektiven, Ed. R. Berlinger
& F. Fink, Frankfurt/ M. 1971; III, p. 131.
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