Number and Logos
Unforgettable Hours with Warren St. McCulloch
(Part 4 of 4)
|The author must confess that
for one reason or another he had forgotten these remarks when he wrote "Natural
Numbers in Trans-Classic Systems". But the memory came painfully back to him when
later on he tried to apply his number concept to Hegel's system of Dialectics. Only then
did he realize that McCulloch's startling statement that a 3 in a system which permits
counting only up to 4 is logically not identical with the 3 in a system where the count up
to 5 is permitted was linked to the fact that even in its own order of numerality a given
number loses something of its rigid identity when the numbers are mapped onto a
many-valued logic. It was obvious that, even by mapping numbers onto a trans-classic
system of logic, they could not change their positions ,"lengthwise". A 3
remained always a 3 and could not move to the place of 4. Thus 1 + 1 remained always 2,
but if the position of 2 was not a fixed point on a, so to speak, horizontal line, one
could always ask: at which locus of the line the 2 was located. Thus, according to the
location, the number could have different meanings. In other words: any number system of
finite length represented itself to a philosopher as a hermeneutical order. Thus even the
number 2 was already open to conceptual interpretation. Seen from here it was obvious that
a system of higher numerality offered more chances of interpretation in a metaphysical
sense and that therefore every time a successor number was added the previous system was
semantically discarded, which meant that each specific world concept had its own numerical
system fitting its own philosophical requirements. If at this stage we use the term
'number' it should be understood that we do not mean what Aristotle calls
,,mathematical" number or "number made of 1 's"
but what we shall call here the
esoteric number following terminological usage in which the lectures of Plato which he did
not write down himself have been frequently called his esoteric doctrine. The
indeterminate duality, e. g. is such an esoteric number. And so is any number which
measures the distance between the universal One () and the last particular
the occasion. It is obvious that the Aristotelian numbers count empirical things or data
of the world we live in and that the esoteric (Platonic) numbers are only concerne with
the realm of Ideas.
Many comments made on the difference between counting in the Aristotelic and the Platonic sense remained very hazy to the author at the time he heard them and he is not certain how much of what he has still to report on the philosophy of numbers is McCulloch's or his own understanding of the problem. It should also be added - and this troubles him very much in retrospect - that in his talks with McCulloch neither ever referred to the concept of a kenogram (5). This has been very annoying to him in two respects: first, in order to get on paper what he had learned from McCulloch on numbers he found it tinavoidable to use kenogrammatic structures and second, since not even the term was ever used, there was not opportunity to ask McCulloch what he made of the difference between numbers within the space of a kenogram and numbers counting the kenograms. Since then, the issue has become extremely important, much more than the author had anticipated in former years, and this again impedes his memories of McCulloch's fundamental philosophic concepts. He is only certain that McCulloch during his last period would have agreed with Klaus Oehler's statement: "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 association of esoteric Number with Idea seems to require another agreement with Oehlers Plato interpretation that esoteric number sequences are completely dominated by the principle of finitude. When we refer in every day life to natural numbers we assume automatically that they form an unending sequence. But if we trust Oehler's interpretation no Platonic system of esoteric numbers ascends an endless way toward the One, nor can it happen that it descends into the bottomless.
Thus peculiar dialectic situation is produced for the earthly thinker. He has the choice of interpreting the Peano sequence of numbers as an ultimate dilution of the orders of esoteric numbers to a degree where they become unfit for the representation of philosophic problems and where they are only good for showing money amounts in cash registers or temperature grades on the scales of thermometers and for similar trivial tasks. But we can also look at them as the material from which we build up orders of esoteric numbers starting from systems with minimal complexity to ever increasing structures of higher order. This produces a scale that proceeds from finitude to finitude! An infinite system of esoteric numbers is inconceivable. If trying to think it we cannot help but apply the numbers of the Peano sequence - which means: we drop out of the realm of metaphysics.
What has just been said is important to elucidate the philosophical radicality of McCulloch's principe of finitude which finally led him to the observation that the finite, metaphysically speaking, is not embedded into an infinite Absolute but that wherever we meet concepts of transcendence the latter will be finite and the Infinite will be is subordinated content.
McCulloch not infrequently remarked that ist was necessary 'to lay the ghost of the Absolute', since in the philosophical tradition the Absolute and Infinity are invariably equated. Heidegger's treatment of the Nichts seemed to him a confirmation of his views. This was very difficult to understand, especially for somebody who was constantly aware of Heidegger's contempt for a thinking that arithmetizes (rechnendes Denken) and who could not forget the severe criticism McCulloch as a psychiatrist had at a different occasion launched against Heidegger and his work. The author was bewildered; but he regained some understanding when McCulloch casually remarked that Peano's definition of a progression, applied to the system of natural numbers, tacitly assumed that we know what Zero is. It was this remark which helped the author very much when, following McCulloch's trend of thought, he developed a system of trans-classic numbers.
In order to make clear how the author tried to implement McCulloch's comment on Zero and Nothingness it will be useful to start with Leibniz' dyadic method of counting:
The left side of Table I displays the sequence of natural numbers expressed in the binary fashion; on the right side we note (always in parentheses) the same sequence in the conventional decimal fashion of writing. If we extend the method of Leibniz to write numbers to a ternary sequence of notation we obtain
Both Tables have two characteristica in common:
a) 0 never turns up in the first place of
a vertical sequence; and
Yet there is a significant distinction between both Tables: since no sequence is permitted to begin with 0 it is impossible that there will ever be structural redundance in Table I; in other words: as long as we stick to two symbols our representation of a Peano sequence cannot be negated, without violating our first rule. Table II shows a different picture. We notice at once that in the group of the two-place sequences (this time written horizontally for convenience' sake) 1 0, 1 2, 2 0 and 2 1 are structurally (morphogrammatically) identical; so are 1 1 and 2 2. In other words: what Table II displays is not a sequence composed of kenograms. This redundance of structural characteristics would also occur in quaternary, quinternary and any subsequent Leibnizian notation of counting.
It stands to reason that in both cases (represented by Table I and II) 0 is given a very specific interpretation: it is assumed a limine that an unlimited supply of zeros is available forming an indifferent background against which numbers can be written. But zero may be interpreted differently.
However, if one attempts to write down with more or less chance of success an adequate representation of the esoteric numbers of Plato one has to abide (using as a mere convention the same kind of symbols) by two principles: first, every number must begin with 0 - as an initial symbolic expression, designated as such and no other symbol may be placed in the notation unless the symbol of counting in our conventional order of signs for counting 0, 1, 2, 3... has turned up at least once. This means that e. g. a fourplace sequence, 0 1 2 1, is a legitimate expression. 0 2 1 1is not, because it only repeats the morphogrammatic structure of the first four-place sequence. It follows that a system of esoteric numbers would have an approximately pyramidic shape and that every horizontal layer would represent a relatively independent numerical system beginning with 0 and ending with the highest number which is structurally permissible in the system.
Peano had used three primitive notions:
Since nought represented no quantity, it was self-understood that his expressions had always to begin with a number denoting a measurable quantity. Nought represented only a boundless background against which numbers could be placed. This meaning of nought, of course, changes, when the distinction between foreground and background becomes irrelevant in an attempt to use a quantitative order of symbols to represent structure. It stands to reason that such a combination of quantity and structure must always have a highest number. And since McCulloch had at least approved of the distinction between iteration and accretion it was always a question how many structural differences can be accomodated between the 0 of accretion and its maximum.
Table III represents an attempt to display a Platonic system of esoteric numbers for a maximum of four places. It is the equivalent of one section of Table VII in Part II of "Natural numbers in Trans-classic Systems". Whether it would have found the approval of McCulloch as a representation of some of his ideas we will, alas, never known.
Table III of this report gives at least an inkling of what McCulloch might have meant with his ruminations that every way to understand the Absolute must be finite; but, on the other hand, Table III also suggests that some caution is needed if we want to reverse the classical thesis that all earthly existence is finiturle and as such encompassed in the infinite Absolute. It is true that whenever and wherever we try to confront the Absolute the face it shows is that of finitude. But Table III also demonstrates that it belongs to the attributes of the Absolute that every finite aspect of it which we discover is followed by an unending sequence of aspects of higher complexity.
At this point an intricate problem of number theory evolves as the numbers which make up the increase of accretion are the esoteric numbers. For the numbers available to us when counting the sequence of the esoteric number systems are the numbers of the non-esoteric Peano order.
How much McCulloch was aware of this ramification of the problem the author does not dare to say. He was hoping to clear that point after McCulloch's return from Europe. He never saw him again. Nevertheless, despite all too many uncertainties about McCulloch's Weltanschauung, the author is convinced that he should be counted among the outstanding philosophical figures of this epoch. Yet it is extremely doubtful whether McCulloch would have been acclaimed as doubtful whether McCulloch would have been acclaimed as such in professional philosophical circles, had he been more outspoken on philosophic issues. His ever deepening conviction that the ultimate key word of philosophy is not Idea but Number is still anathema in the departments of philosophy as well as in the Humanities. The author himself confesses that if somebody - before he had the good fortune of knowing McCulloch - had suggested that in Metaphysics we require numbers in order to understand ideas instead of saying that ideas are necessary to understand numbers he would have more or less politely changed the topic. It took a McCulloch to show him that it had been the tragic fate of Western civilization to permit the concept of the idea to gain metaphysical precedence before number and that from this very choice the fateful split between sciences and the humanities had resulted. In McCulloch there was no such split. In the eyes of the author this courageous reversal in the order of idea and number alone makes him a philosopher of most impressive stature. It is impossible to measure the philosophical import in detail because this is a matter of future historic developments. For the time being the traditional viewpoint prevails overwhelmingly. But one may safely say that his work and the philosophic attitude underlying it has created the conditions for a total reversal of the logical foundations in the humanities, and it has set a standard for future cybernetic work. The author has never concealed his dissatisfaction with the pitiable paucity of guiding principles metaphysical in the pursuit of cybernetics. Only after McCulloch's death has he been told that he shared this dissatisfaction and did so with an equal degree of intensity. He was aware long before the author that cybernetics was not just a novel technical discipline among others but that its future pursuit implied a new philosophic concept of reality. Fundamentally it is nothing less than a new form of philosophic thinking under the guise of a particular scientific discipline because it endeavors to give to the philosophic method, via neurology and related fields, a precision it had never had before.
A short report of certain consequences of McCulloch's thinking on a domain remote from cybernetics may illustrate its philosophical relevance.
It is the area of philosophical hermeneutics as applied in history and other branches of the humanities. For the time being it seems absurd to approach hermeneutics as Dilthey and his successors understood it with arithmetical procedures. A number is always what it is, and the result of an arithmetical operation is either true or false - or undecidable. There is not the slightest room for 'interpretation'. But if we look at the numerical system evolved in the manner in which Table III demonstrates it is no longer enough to say: This is 2, this is 3, this is 4 etc. Because even if we add 1 to 1 equals 2, the question already will haunt us: which 2 do you mean? 2 in the iterative, or 2 in the accretive sense? If we read Table III from top to bottom there is no case in which a number has just one successor; it has at least two mostly, however, more. In Table III the fully aecretive version of 4 would e. g. have five successors. In order to obtain this situation nothing has been done but apply the elementary dichotomy of sameness or otherness. This has the effect that, beginning with 0, an ever increasing amount of Peano sequences of non-esoteric numbers are spreading out in different sequences of esoteric numbers. However, as far as a given system of esoteric numbers is concerned the principle of successorship is not the one which we have just describes. In these finite number sequences which we have to read horizontally every "esoteric" number has just one and only one successor - except the last which is fully accretive; it has therefore no successor at all. Correspondingly, the first, which is fully iterative, possesses no predecessor. It follows that the principle of hermeneutics originates only the transition from one finite system to the subsequent one with increasing structural properties. But as long as we remain on a given esoteric level the principle of single successorship holds unconditionally.
If we want to express ourselves in
Platonic terms we may say that the esoteric numbers
It had to be so. When Rufus Jones, the Quaker, asked him in his youth what he wanted to do in his life, he told him that the guiding star of his thinking would be the question of numerosity. When the author met him in the evening hours of his life McCulloch had remained true to the self-dedication of his youth.
The reference to the Platonic numbers might suggest that McCulloch was basically a Platonist. However, such judgment would be far from the mark. He was well aware that Platonism in its narrow sense belongs to an epoch of philosophic thought which had seen its heyday. For him philosophy still oscillated between two fundamental inquiries.' is reality rooted in a last irresolvable discord or in a final coincidence and reconsiliation of all contradictions? The "Embodiments of Mind" give the impression that he leaned more toward the concept of a final resolution. In the "Mysterium Iniquitatis" we read that "cybernetics has helped to pull down the wall between the great world of physics and the ghetto of the mind" and "so we seem to be groping our way toward an indifferent monism". But the author, during the very late sixties, heard sometimes statements which were not exactly in accordance with the last quotation. The author remembers one occasion when McCulloch attacked psychoanalysis with a degree of animosity and the author drew his attention to a short sentence in the "Past of a Delusion" where he had read: "Upon Causality herself Karl Marx begat his bastard, Dialectical Materialism." The author who never considered himself a Marxist but an Hegelian stoutly defended Dialectics (and never mind the distinction between dialectic idealism and dialectic materalism). For him any transcendental theory of the universe had to have dialectic structure McCulloch denied the validity of this position but he was interested enough in the issue that some sort of discussion ensued. In its course he developed some ideas which fitted in ill with his leanings toward monism. The author is not sure whether they expressed some real convictions and new philosophical insights or whether they were merely argumentative stratagems to win over his opponent and disabuse him of dialectics. The author is inclined to believe the first: but he is by no means sure about it.
McCulloch casually referred to the Buddhistic Nirvana and insisted that European concepts of Reality were too deeply associated with the idea of ,,Substance" at the expense of "Relation". As always when he talked with the author he drew his exemplifications rather from formal logic and abstract number theory and not from cybernetics proper. Commenting on his suspicion that the concept of substantiality played too large a role in Western philosophy at the expense of the problem of relationship he speculated what philosophy would look like if we stopped talking so much about ultimate building blocks of the Universe and postulated that there were no such things and that every assumed last unit was nothing but a relation of even more fundamental units and that this splitting of the building blocks was a process that could never end. As a firm believer in dialectics the author could only agree. It fits in quite well with McCulloch speculations about numbers and Finitude. On the other hand, his musings on Substance and relation do not harmonize with the concept of an "indifferent monism" because there is no transcendental 'space' in which the difference between relator and his relata may ever disappear (6).
Unfortunately, there remains a rest of doubt. McCulloch showed as usual an extraordinary reluctance to criticize the arguments of his opponent and to reveal much of his own philosophic forays into the Ultimate.
One thing seems certain, however - the philosophic position displayed in the "Embodiment of Mind" does not fully reflect what McCulloch thought during the last years of his life. He was no longer certain - as we still read in "Through the Den of the Metaphysician" - that "the seeming contradictions vanish in the grace of greater knowledge". His concept of metaphysics had deepened and he frequently made statements that were difficult to reconcile with the remark in the "Mysterium Inquitatis of Sinful Man" that notions are metaphysical if "they prescribe ways of thinking physically about affairs called mental". Many of his remarks daring the very last years would have suggested that by metaphysical terms he understood concepts which refer to a situation in which it was on principle impossible separate object and subject, including the thinker.
The author is led to this conclusion by McCulloch's reflections on the mutual logical position of Substance and Relation. There is no way in which Relation can ever be dissolved in a term of substantiality and vice versa. On the other hand. a relator and its relata depend functionally on each other, neither makes sense without reference to the other. They are - as Hegel would say - dialectically connected, and the problem of this connection defines the realm metaphysical. The author believes that McCulloch might lastly have agreed. If one shifts from the distinction between 'physical' and 'mental' in his former definition of what he would be willing to call "metaphysical" to the radically logical contrast between relation and relator it is obvious that the meaning of the term 'metaphysical' must also change. In the sense of Hegel's logic the distinction between relator and relatum can never "vanish in the grace of greater knowledge". While only relata may designate substance metaphysically the relator refers for ever to an act of subjectivity. This requires a deeper insight into the philosophical problem than cybernetics possesses at the present moment.
When the author was told that McCulloch was seriously dissatisfied with the development of cybernetics he could well understand it. But while writing this essay and trying to trace McCulloch's philosophic reflections into greater depths he has also learned to understand his reluctance to criticize the turn cybernetics has been taking. In his last years he was experimenting with new thoughts but had not reached the degree of certainty where his scientific conscience would have permitted him so speak aloud of his doubts and misgivings.
It might be possible to draw a clearer picture of McCulloch's last philosophical reflections; but this would require a greater amount of interpretation by the author - in other words: it would have been progressively more difficult to distinguish between what McCulloch had been thinking and what the author thought he did think. For this reason greater clarity and coherence has been sacrificed to the aim of at least approximate historical accuracy. The author is sure that he has not succeeded in the desired degree. He only knows that apart from Plato, Aristotle, Leibniz, Kant and Hegel - no modern philosophical thinker has exerted a greater influence on him than Warren McCulloch whose memory he shall always cherish and revere.
(5) Except in a
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