Number and Logos
Unforgettable Hours with Warren St. McCulloch
(Part 3 of 4)
|With this thesis that not
the Finite is embedded in the Infinite but that the Infinite - be it conceived as
potential or actual - is, in the metaphysical sense, only a subordinated element of
Finitude McCulloch showed himself to be a first rate metaphysician. This view of
Metaphysics had never occurred to the author though he had always prided himself of having
effected in his: "Cybernetic Ontology ..." a metaphysical breakthrough from
classic tradition by means of the rejection value. But McCulloch went much farther with
his reversal of the mutual role of Finitude and the Infinite. Whenever classic tradition
through the history of Philosophy discussed the meaning of the Absolute a philosopher
would have deemed to have lost his senses if he had proclaimed that the Absolute is a
Finitude and that the main characteristic of the empirical world is its Infinity.
Unfortunately, McCulloch did not elaborate this point in detail. And the author did not
press him very much because he hoped to have, later on, a better occasion to elicit a
detailed explanation of this startling and paradoxical theorem. Alas, this oppurtunity
There was just a hint of an explanation in his evaluation of the Platonic confrontation of the One and the , the indeterminate duality. He approved of Aristotle's opinion that this duality was nothing but a material constituent. To put it differently: a number is an entity which is produced by the actual determination of determinable potentiality. And the vehicle of the determination is always the One. McCulloch agreed with this Aristotelean interpretation but not wholeheartedly. He told the author again and again that this way of thinking overlooked something and did not account clearly for the difference between the step from 1 to 2 in the familiar sense of Peano sequence and the step from Oneness to Duality in the other sense that Duality already implied an unbounded manifold. It had been noted before that Aristotle seems to be confused about the difference between the "indeterminate duality" and the number 2 (A. E. Taylor; Plato, N.Y. 1927, p.512); knowing this McCulloch's arguments gained a greater weight with the author than they would have done otherwise. He decided, startled by the novel metaphysical viewpoint of McCulloch, to attempt a new interpretation of natural numbers on the basis of a many-valued logic with a kenogrammatic background. He sought and obtained McCulloch's agreement not to follow the way of Barkley Rosser but to choose a different method. There was nothing in Rosser's paper on undetermined duality, whereas McCulloch and the author agreed that the meaning of this term was the key to the whole problem. Aristotle's lack of the understanding of the problem led to a position where he could only recognize what he called 'mathematical number' which is nothing but what we have called Peano numbers. The other numbers, the numbers of Platonic ideality, which define the Platonic order of ideas would not possess any logical legitimacy if we wanted to follow Aristotle. This, according to McCulloch, was unacceptable because the order of the Peano numbers was intrinsically incapable to reproduce the conceptual wealth of the system of Ideas. In McCulloch's opinion Rosser was still and Aristotelian in his number theory. When the author, with some trepidation, decided to leave pure logic for the time being and tackled number theory he was warned from some other side that his lack of mathematical training could only lead to an abysmal failure. With his first sketch which he called proto-numbers he went to McCulloch and told him of the warning he had recived and made no bones about his mathematical incompetence. However, he was at the same time able to point out that the same argument could have been applied to the corresponding efforts of the mathematicians. Since Frege there had been strenuous efforts to give mathematics safe logical foundation but it could hardly be denied that the logic underlying these efforts nowhere went beyond Leibniz at best and that neither the transcendental turn effected by German Idealism nor the problem of dialectics and its destinction between Platonic and Hegelian dialectics was properly understood on the side of the mathematicians. Here stood incompetence against incompetence and it could only be hoped that a better cooperation between mathematics and philosophy would produce something worth while. McCulloch encouraged the author to continue who took it as part of the encouragement that McCulloch invited two or three friends and collaborators of his to whom the author should present his ideas. He has now forgotten who else attended but he remembers that Professor Manuel Blum was present. Taking into consideration everything McCulloch had said about the indetermined duality and also including the result of discussions on Hegel the author took the following step toward a transclassic theory of natural numbers. Guided by Hegel's dialectics he said that the process of adding 1 to a preceding number was ambiguous: it could either be interpreted as "iterative" or as "accretive". Starting from 1 and proceding to 2 the duality thus obtained was indeed indeterminate but not in the sense which Plato, according to his interpreters, might have intended. Interpreters have usually been of the opinion that for Plato going from 1 to 2 was only the step from Oneness to Manifoldness and that the indeterminacy of the manifold which this step established was not positively fixed. It could be anything: 2, 3, 4 and so on.
The argument against this interpretation is that it does not lead to dialectics and Plato was a dialectician. His doctrine of ideas clearly shows a dialectic structure and if the order of the ideas is determinable by numbers then the numbers themselves must display a dialectic structure also. This was a consequence McCulloch had not only admitted in the discussions with the author. More so: he had pointed it out to him before the latter had become aware of it. The dialectical treatment of natural numbers - 'dialectic' in the combined meaning of Plato and Hegel - implied that the process of addition 1 + 1 = 2 should be interpreted in two ways: one could either look at the two 1's as being identical or as being non-identical. This could be done by either ignoring the fact that the second 1 was a repetitor of the first 1 or by not ignoring the repetitional character of the second unit. The result is different in both eases. No matter which interpretation was chosen the result would, of course, always be a duality. But duality would carry two meanings; it was important to express this in a way that the difference in meaning would become computable.
At this point the author was helped by a stray remark McCulloch had made a year ago the importance of which the author had previously overlooked. McCulloch said that the difference of meaning seemed to him a difference of quality in the sense in which Hegel differentiated at the beginning of his Logic between Being and Nothingness as antithetical qualities. Only in this way could one understand how dialectics might finally turn qualities into quantities. The author found this remark extremely cryptic and asked McCulloch how this dialectic transition might happen. He got the disappointing answer: This is for you to find out. At a renewed attempt to extract at least some shreds of information pertinent to the problem the author was only reminded of a former discussion about Heidegger and his treatment of the Nichts (3). This he considered no help at all. But then he found his attention drawn back from the concept of number and directed towards the idea of the kenogram. Kenograms are empty places which may or may not be occupied by values. Up to this point the author had always believed that only one value at a time could occupy a single kenogram. Not it occurred to him that a kenogram might behave differently in the ease of numbers, and that it might be the ontological locus not just for a single number but for a total Peano sequence of natural numbers. And since a Peano sequence is of infinite extent such numerical order would be a demonstration of McCulloch's startling metaphysical thesis that not the Finite is encompassed in the Infinite but that all Infinity must be understood as a subordinated element of Finitude, i. e. a kenogram. The author was so excited by his brainwave that he did what he had never done before and as far as he can remember never did afterwards, he rang McCulloch up to ask his opinion. Contrary to his expectation McCulloch was not swept off his feet but asked all sorts of question how a single kenogram could be defined as an all-encompassing domain accomodating a never ending process of counting. There was nothing in the original conception of a kenogram, so McCulloch reminded the author, that would suggest such property. The author must confess that he felt deflated when he hung up. But his respect for McCulloch's mental acuity was so great that he settled down immediately to think the problem over. Very soon his initial disappointment turned into deep gratitude, because out of McCulloch's critical remarks the concept of the universal contexture was born. The author is convinced that he would never have found this idea if he had not been privileges to listen to McCulloch's thoughts about the metaphysical rank of Finitude and the information given over the telephone. Re gratefully acknowledges that McCulloch is as much the creator of the concept of universal Contexturality as opposed to mere context as the author of this essay. For this reason it seems to be fitting to describe here the difference between a mere context and a universal Contexture.
If, e. g. in court the question is raised whether the defendant is guilty or not guilty, it would be non-sensical to answer: no, he is broad-shouldered. In other words: the alternative guilty or not guilty is enclosed in the context described by the statutes of criminal law. On the other hand: the question: 'Is the growth in this person malignant or non-malignant?' cannot be answered by: 'No, he is a poet', because the alternative which has been raised belongs to the context of pathology. In both cases the answer must be guided by a tertium-non-datur which refers to a superordinated viewpoint which in our first ease was criminal law and in the second pathology. The alternates of a context may be very narrow and again they may be of ever increasing generality, the alternative still constitutes a mere context as long as it is possible to determine a superordinated viewpoint. A context changes into a universal contexture only on condition that it is impossible on principle to find a superordinated viewpoint which defines the meaning of the tertium-non-datur for the opposites for which the superordinated common viewpoint has been sought. The classical example for this situation is Hegel's "alternative" between Being (Sein) and Nought (Nichts). They are alternatives which exclude each other. Nobody can deny it. Yet nobody can conceive of a metaphysical concept that would be of greater generality than both of them. In other words: both constitute seperate universal contextures. We are not able to understand the distinction between Sein and Nichts as alternatives within a context. The question: of what context? must in this ease remain unanswered. Similarly we read in Lenin's works that for the opposition of Mind and Matter no common denominator of higher generality can be found. Mind and Matter are not elements of a context. They are universal contextures, capable of encompassing contexts with limited alternations. Lenin concludes from this insight that the thinker who has arrived at this alternative has come to an end of his theoretical way. He is only left with the decision to declare himself either an idealist or a materialist. This is not the place to sit in judgment of the legitimacy or illegitimacy of Lenin's conclusion but his example shows that the situation Hegel discusses at the beginning of his Logic can turn up under radically different aspects (4).
If the reader thinks that these reflexions are far from what we read in the ,,Embodiments of Mind" he may be reminded of the insight the essay "A Heterarehy of Values determined by the Topology of Nervous Nets" conveys. There we learn that ,,an organism possessed (at least of six neurons) is sufficiently endowed to be unpredictable from any theory founded on a scale of values.. It has a heterarchy of values, and is thus internectively too rich to submit to a summum bonum."
A summum bonum requires an ultimate hierarchy of values with an absolute value at the summit. Logically this means that there must be a tertium-non-datur crowned by a final common denominator of 'Sein' and 'Nichts'. If somebody insists that such a denominator is inconceivable the hierarchist will willingly agree but explain that this ultimate common denominator is nothing but God himself, as the Lord of a monocontextural Universe. McCulloch's heterarchy of values, on the other hand, postulates a reality that is only conceivable in a poly-contextural sense. In other words: the world we live in cannot be understood as an unbroken universal context. In fact, the term 'universal context' is in itself a contradictio in adjecto. It may be true that the author finally formulated the difference between context and contexture, but it is also true that he could never have done it without the spade work McCulloch had provided.
In fact, there is another way to show how near McCulloch came to develop the distinction between context and contexture. He had an amazing knowledge of medieval logic and he once referred to the famous ninth chapter of Peri Hermeneias and its influence on medieval logic up to William Occam. Aristotle had stated that in logical terms the difference between Past and Future could he defined by the fact, that the tertium-non-datur is valid for and applicable to all the Past. With regard to any Future the tertium-non-datur is equally valid, bui it is not opplicable. McCulloch considered this distinction very important for the understanding of the present, and it shows how near he came to distinguish between context and contexture because, if we refer to the Past, we refer to what has happened in a context. Thinking about the Past we always mean the actual contents of a contexture, thinking ahout the Future, however, we can only refer to an as yet empty universal frame which has not yet been filled with any contents hecause, if it were, it would not be the Future. Writing down these lines the author wonders how far he is perhaps plagiarizing McCulloch. Because he is convinced that his own thoughts might not have gone in this direction if he had never had the good fortune to have those long nocturnal talks with McCulloch.
It was not always easy to listen to him, because his way of thinking was seasoned, as Seymour Papert rightly remarks, "with a very personal flavor" which not unfrequently led to misunderstandings. One example was his pronunciamento that Finitude should he given metaphysical priority over the Infinite. The author is by no means sure that he has caught the full meaning of what McCulloch really intended by this statement. It is much too simple an assertion to describe an involved situation correctly. But it was one of the suggestions which helped him to arrive at his own distinction between a contexturality and its potential contents. A universal contexture is a finitude insofar as it is only one piece in a patch-work of an unbounded multitude of contextures. It is limited by its borderline to a neighboring contextural domain, but its capacity for content is unlimited owing to the peculiar character of its tertium-non-datur. When talking about the metaphysical priorities of finitude and infinity McCulloch casually mentioned Heidegger's "Seinsvergessenheit". If the author understood him properly - which is by no means certain since the morning was dawning and he was overtired - then Heidegger's "Seinsvergessenheit" must not be understood as a term referring to the contexture 'Sein' but to its contents only. On the other hand, when the talk focussed on Heidegger's 'Nichts' it was a foregone conclusion that the contextural frame was referred to, because it would have been nonsensical to speak of the actual contents which nothingness might encompass.. Further, it must be understood that the expression 'universal contexture' was understood that the expression 'universal contexture' was not used either by McCulloch nor the author at that time because neither was ready for it. Instead of it rather involved circumlocutions were used. However, trying to distill from his memory what seems to him the essence of the discussion the author finds it easier to use this more precise term which assuredly was a result of the mental exchanges between McCulloch and the present reporter.
During the last meeting the author had just returned from his yearly skiing vacation - it was agreed that he should write a paper on natural number theory within the frame of trans-classic logic for the next meeting of the ASC in Gaithersburg. The author remembers he had grave doubts that his paper would be ready for the third Annual Symposion of the American Society for Cybernetics. In consequence of his misgivings he informed McCulloch that he did not yet know whether he would be able to offer something in time to the Society. It turned out later that his pessimism was unjustified and he completed within the deadline the second part of the text which later appeared in the July/September issue 1971 of the Journal of Cybernetics. McCulloch did not know it; he had been in Europe during this period and when he returned he asked Dr. Edmund Dewan whether the promised paper had been handed in. This the author was told by Dr. Dewan on the first day of the Symposion which McCulloch could not attend because he had died on Sept.24, 1969 in Old Lyme, Conn.
When the paper was finally published with a Part I preceding the original text now designated Part II the writer added a footnote that the ideas expressed in the first part were to a great extent the result of a night session he had with McCulloch toward the end of February 1969. Since then 5 years or more have past. and his memories of McCulloch have gained a new dimension. He knows now how much more he owes to McCulloch than this footnote expresses. The maturing of his memories has shown him among other things that McCulloch's influence did not only extend to one part of the aforementioned essay but to the other part as well. It was one of the remarkable gifts of this great man and scholar that he developed in his associates ideas and mental trends which they themselves might never have brought to fruition unassisted. The author of these remembrances has endeavoured to show how McCulloch, by delving deep in the philosophic aspect of Finitude elicited from the brain of his listener the conception that the Universe we live in is not mono-contextural but a network of Finitudes, partly bordering, partly overlapping, and in the case of compound contextures even encircling elementary contexturalities, in short: a polycontextural Universe. He deeply regrets that McCulloch never saw the final text in order to give or deny it his imprimatur. He feels that the philosophical impact of McCulloch's thinking is still vastly underrated even by his admirers an disciples. He was such a many-sided thinker that he appeared enigmatic, never showing all facets of his mind to a single partner in discourse. To a neurologist he was an innovator in neurology: to a psychiatrist he revealed new ideas on psychiatric problems; with a mathematician he would discuss the mathematical aspects of his work, and when he met the author it was in the den of the metaphysician.
The quantity of topics McCulloch liked to talk about was enormous and his roving mind led the listener, sometimes quite unexpectedly to connections which went far beyond conventional associations. But wherever he turned to the problem of ultimate or penultimate foundations he looked for his datas in the realm of numbers and number was for him invariably linked with Finitude.
Once the general topic of discussion had been a passage in "Why the Mind is in the Head?" concerning the relation between quantity and number. There we read that 2 in so-called analogical contrivances a quantity of something ... is replaced by a number... or, conversely, the quantity replaces the number." When the author suggested that, following the example of Hegel's Logic, the triadic relation between a quantity, number, and quality would also deserve a closer look, McCulloch switched to the question: why in primitive societies the capacity of counting was often very limited. The most elementary system of counting would, of course, work only with three hazy concepts: oneness, duality, and general manifoldness. McCulloch insisted that something was conceptually wrong when Plato according to tradition included general manifoldness in the concept of duality only because duality was not longer oneness. This improper inclusion was due to the fact that classic logic permitted onlv two values and nothing beyond. But then McCulloch continued that, if a finite system of numbers increased by the addition of one more numerical concept it would no longer be the same system to which a new numerical unit had been added, but it would be, logically speaking, in its totality a new system of counting! And every time one more unit was added this was not an adding process in the conventional sense in which we increase a given quantity by adding just 1. Instead, by addition we abandoned the numerical representation of a given conceptual order and moved to a different conceptual relationship with a somewhat higher complexity. This means that - let us say - the number 3 in a numerical order that went up to 4 was logically no longer identical with the 3 that occurred in a system which permitted you to count up to 5. To melt all these logically distinct systems of finite counting together into an unending Peano sequence one had to suppress most of the logical distinctions which number as a metaphysical concept implied. For this very reason number as a medium of thought had fallen into disrepute in ontology and was forced to make room for conventional language to represent metaphysical concepts.
(3) See also: Martin
Heidegger, Was ist Metaphysik? Frankfurt/M. 1951, pp.22 to 38.
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