Number and Logos
Unforgettable Hours with Warren St. McCulloch
(Part 1 of 4)
|The author of these
remembrances (from now on only the 'author') feels painfully that he is in an awkward
position. He intends to show a side of Warren McCulloch which is not very well - if it all
- known and which hardly becomes visible in the publications of this very great man and
first rate scientist: we refer to his importance and profundity as a philosopher. He was
aware and very intensely so - of Cybernetics as a discipline sui generis that needed a
novel philosophic foundation to distinguish if from the conventional disciplines. This
conviction of his finally led to the meeting with the author - a contact which lasted
almost a decennium. The quandary the author finds himself in stems from the fact that he
entertained and still entertains almost identical views about the relation between
cybernetics and philosophy as McCulloch and finds it therefore almost impossible to
perform a clean separation of his own ideas from those of McCulloch. He is only sure that
the thoughts he expressed on cybernetic topics are fully his own up to the publication of
his "Cybernetic Ontology and Transjunctional Operations" which came out in 1962.
Although McCulloch is already quoted in this essay it was done solely with the intent to
appeal to his authority for ideas which the author had entertained for quite a while.
The contact between the author and Warren McCulloch was established after Dr. John Ford, then at the George Washington University, had given McCulloch in 1959 a German paper of the author "Die aristotelische Logik des Seins und die nicht-aristotelische Logik der Reflexion" which had come out in Germany in 1958. He is still intensely grateful to Dr. Ford for having made this connection which was bound to change his total outlook on philosophy. However, it took some time before he really understood what had attracted Warren McCulloch to his paper. It was not so much its potential applicability to cybernetics but a hidden relation that it revealed between number and logical context. When the author wrote it he opined that a non-Aristotelian Logic is nothing but a place value system of innumerable logical sub-systems of Aristotelian (two-valued) character. His interest was at that time wholly conceptual and he did not even dream that a hidden arithmetical issue might lead into deeper foundational layers of Cybernetics. Here McCulloch was far ahead of him.
Their intellectual collaboration started in earnest when some evening the author had made a stop-over on his yearly trip to New Hamsphire - McCulloch led the talk to the Pythagoreans and their theorem that numbers describe the ultimate core of Reality. Although the author pressed for a detailed explanation all he was told at that time was that to find out more was exactly his own business. It was the first time that the author encountered a peculiar reticence of McCulloch's regarding ontological or - more precisely - 'metaphysical' questions. It led him to grossly underestimate McCulloch's gifts and intuitions in this direction. He was confirmed in his faulty judgment when he noticed that McCulloch never bothered to make corrective remarks when a paper which was read at a congress or sympo- sion where he was present obviously implied metaphysical assumptions which had to be partly or totally wrong. First he assumed that McCulloch was not aware of it; later however the author knew better. Nevertheless he must confess that during the whole duration of his acquaintance and - as the author hopes friendship McCulloch never gave up his reluctance to criticise the course cybernetics was taking with relation to Philosophy. Only after McCulloch's death he learned that his mentor in Cybernetics had been as dissatisfied as he himself with the lack of fundamental ontological orientation that characterized - and still characterizes - the pursuit of cybernetic theories. But he came to understand very soon how much McCulloch saw his own endeavours within a novel metaphysical frame. The revelation came one evening when McCulloch started to talk about Martin Heidegger and produced a copy, very shabby and dilapidated from intensive use, of "Sein und Zeit".
The book had originally belonged to his friend and coworker Eilhard von Domarus, so he explained; he in his turn had studied it carefully and he now wanted to give it to the author for renewed study because the latter had confessed that he did not care very much for Heidegger's philosophy. The expression of thanks for the unexpected present must have sounded rather reluctant because McCulloch grew very eloquent and insisted that the "Nichts" (Nought) in Heidegger's philosophy was precisely the ontological locus where the central problem of cybernetics was located, namely the mapping of the process of Life onto matter per se inanimate. BEING is both: subject and object as well; but western philosophy has fallen into "Seinsvergessenheit" (oblivion of ultimate Reality) since the time of the Greek. Which in McCulloch's view meant: it did not focus on the problem of cybernetics. In classic philosophy mere objectivity without self-reference is mistaken for "Sein". When McCulloch commented on Heidegger with these remarks the author knew he had underestimated his philosophical gifts. His detailed knowledge of "Sein und Zeit" and especially his discussion of this "Nichts" gave the author's metaphysical thinking a new direction and made him look for the roots of Cybernetics in the ultimate and primordial recesses of the Universe.
Since the spiritual contact point between MeCulloch and the author happened to be their common interest in the transcendental relevance of logic in other words: how much and what information logic conveys about the world that surrounds us - it was only natural that the author wanted to know from his partner what he meant by the term 'metaphysical'. For a start he was referred to the "Mysterium Iniquitatis ..." and the notions that "prescribe ways of thinking physically about affairs called mental ..." It stands to reason that this answer left the philosopher dissatisfied and it surely did not cover McCulloch's own - very ambivalent appreciation - of Heidegger. This was admitted; and then MeCulloch started to express thoughts which went far beyond the metaphysical references imbedded in papers like the "Mysterium Iniquitatis" "Through the Den of the Metaphysician", "What is a Number ..." and others. He drew the author's attention to the fact that any logic or calculus Man may ever conceive is nothing but a more or less competent formalization of ontological concepts. This ideas was, of course, not new and may be easily extracted from his writings as ever present implication. But it showed that he had wandered much deeper into the grottoes of metaphysics than he was inclined to express explicitly in his papers. At this juncture the author thinks it fitting to remind the reader of the quotation of Clerk Maxwell appearing in "Through the Den of the Metaphysician" about the relation between thoughts and the molecular motions of the brain: "does not the way to it lie through the very den of the metaphysician, strewn with the bones of former explorers and abhorred by every man of science?" McCulloch comments this quotation with a "Let us peacefully answer the first half of this question 'Yes', the second half No', and then proceed serenely."
While there can be no doubt that he never abhorred the den of metaphysics his texts show a pronounced reluctance to analyze in detail the accoutrements of Transcendence. On the other hand, this reluctance disappeared almost completely when speculating on the pertinent issues in the presence of a person who was much more at home in the realms of the Transcendental than in the empirical ways of Cybernetics as happened to be the case with the author.
From Heidegger's "Nichts" the discourse went to Kant and Hegel. The author must confess that he was somewhat surprised when he discovered that McCulloch understood that Kant's philosophy closes an epoch of philosophical thought and that Hegel opens a new one. He knew this, of course, himself, - that was after all his business - but he had interpreted it in terms of the distinction between 'Natur- and Geisteswissenschaft' and the pseudo-systematic development of the latter in the Hegel-Renaissance since 1900. Of the Hegel-Renaissance and its concomitant intellectual events McCulloch was hardly aware. Even if he had been familiar with it: the metaphysical gap between matter and mind or subject and object which was emphasized by the Geisteswissenschaft could not be accepted by any cyberneticist, least of all McCulloch. Consequently, he explained the distinction between Kant and Hegel by pointing out the different view of Dialectics entertained in the Critique of Pure Reason and in Hegel's Logic. Kant deals with Dialectics in the sense of the Platonic tradition and in the Critique of Pure Reason the dialectic argument ends in the transcendental illusion as the unavoidable admixture of error that infiltrates all metaphysical assertions. Thus Kant's evaluation of Dialectics is basically negative and the less we imbibe of this poisonous drink the better off we are. For Hegel, on the other hand, he explained, the dialectic structure is a legitimate element of thought as well as of objective existence and it furnished the transcendental link that connects both. Seymour Papert has referred to this situation when he reports in his Introduction to the Embodiments of Mind that McCulloch insisted "that to understand such complex things as numbers we must know how to embody them in nets of simple neurons. But he would add that we cannot pretend to understand these nets of simple neurons until we know - which we do not except for an existence proof - how they embody such complex things as numbers. We must, so to speak, maintain a dialectical balance between evading the problem of knowledge by declaring that it is 'nothing but' an affair of simple neurons, without postulating 'anything but' neurons in the brain. The point is, if I understand him well, that the 'something but' we need is not of the brain but of our minds.. namely, a mathematical theory of complex relations powerful enough to bridge the gap between the level of neurons and the level of knowledge in a far more detailed way than can any we now possess." (p. XIX)
After the author had read this introduction he asked McCulloch whether he really intended to introduce dialectics only in a loose and logically non-coercive manner or whether he realized that Hegel employed the term as a linguistic cover for a hidden exact mechanism which the Universe as a whole employed but which we were still incapable of unravelling. McCulloch remained silent for a few moments and then asked the author to rephrase the question, which the latter did by simply inquiring whether he thought that the term 'dialectics' merely referred to a quirk or weakness of the human mind or whether it indicated an intrinsic property of Reality. This time McCulloch answered that the term should designate an objective quality of the universe and he added: I think this is what separates Kant from Hegel. The author and McCulloch agreed that the "so to speak" in the lengthy quotation above was not a proper expression because it suggested only a vague analogy. It did not indicate that in the term "dialectical" a very precise systematic foundation problem of mathematical theory was at hand.
The author cannot now remember how the talk got to a paper of Barkley Rosser "On Many-Valued Logic", which was published in the American Journal of Physics (Vol.9,4; pp. 207-212, 1941), and from there to the question whether a dialectical analysis of natural numbers might help to bridge the gap between the level of neurons and the level of knowledge which is conveyed by present mathematical theory. Everything was still very vague, and it took an almost nightlong discussion to clear the realm of discourse somewhat. It helped greatly that McCulloch was familiar with the distinction of number by Plato and Aristotle and how much nearer to the Pythagoreans Plato's ideas were than those of Aristotle. And then he surprised the author by saying that, what Hegel meant by number was a not very successful attempt to rebuild again the general concept of numerality which had been divided by the antagonism of Platonic and Aristotelian philosophy. He finally added that Hegel failed to develop a novel theory of mathematical foundation because he thought more about number in the Aristotelian than in the Platonic sense. This was a most astounding conclusion and seemed questionable to the author. He believed that he knew more about Hegel and felt unable to accept McCulloch's thesis. Since the whole history of mathematics from the Greeks to the present time owes all its success to the instinctive acceptance of the Aristotelian way of thinking ahout numbers McCulloch had to be wrong. The author left Shady Hill Square somewhat dissatisfied and went skiing.
Six weeks later he was back, very contrite and humble. He was not a mathematician, only a logician, moreover reared in the atmosphere of the Geisteswissenschaften. But it had, in the meantime, dawned upon him how much hetter a philosopher McCulloch was when the mind turned to the problem of the transcendental relation between mathematics and the Universe. Conceding McCulloch his Hegel interpretation the discussion doubled back to the essay of Barkley Rosser. Rosser's attempt seemed now extremely interesting; Rosser had demonstrated in his paper, that one can get numbers from four ideas in two-valued logic which have been formalized in terms of a likewise two-valued calculus. The first idea is 'conjunction' (... and ...); the second idea is 'negation' (not ...); the third idea is 'all'; and the final idea is 'is a member of'. Rosser then suggests a projection of these ideas onto the structure of a many-valued calculus. For the purpose of demonstration and to retain a comparative simplicity he exemplifies his case with a three-valued logic. As values he chooses 'true' (T), 'probable' (?), and 'false' (F). McCulloch and the author agreed that this interpretation of three-valuedness has proved its usefulness in cybernetics and elsewhere but that it could not lead to a trans-classic theory of natural numbers because it has been established since at least 1950 (Oskar Becker) that the introduction of probability or modal values destroys the formal character of a logical system. For if strict formality is insisted on any such spurious many-valued system reduces itself automatically to a two-valued calculus. In order to convince McCulloch that Rosser's approach to the problem needed a weighty correction the author pointed to something which he considered Rosser's second mistake. The latter determines conjunction in classic logic by the following matrix:
and the stipulation that T is not permitted to re-occur in any of the empty places which originate if we extend the places for the functional result from 4 to 9. Thus he defines, in strict analogy, three-valued conjunction by the matrix:
We repeat: in order to retain the meaning of conjunction T is not to go in any of the empty places which are left open in the above matrix. However (?) and (F) may go indiscriminately in any of the other squares. Since 8 squares are left to be filled and since two choices are available in the case of each square there are 28, i. e. 256 possible choices for filling the squares. in Rosser's opinion all of them represent the general meaning of conjunction in a three-valued logic. This claim was easily refutable if one recognized - as McCulloch did - the interpretation of trans-classic logic as given by the author in his "Cybernetic Ontology and Transjunctional Operations". In order to demonstrate Rosser's too generous interpretation of conjunction the author filled out the matrix in the following way:
In order to avoid the ontological consequences which are implied in Rosser's use of the symbols T for truth, ? for probability or modality, F for false we have denoted the values in the same order with the first three integers. This choice of values is quite in accordance with Rosser's stipulation for the meaning of conjunction. However, there it not the remotest chance to interpret this arrangement as a matrix of a conjunctive functor. To render a minimum sense of conjunction a three-valued logic would have to retain the structural feature of conjunctivity in at least one of the two-valued alternatives 1 or 2,2 or 3, or 1 or 3. This is not be case, because or the two-valued system encompassing the first and the second value we obtain the morphogrammatic structure which can only be filled by trans-junctional value-occupancy. For the two-valued system constituted by 2 and 3 we obtain a morphogrammatic structure for value-occupancy which is demanded in the case of equivalence, and for the final two-valued system the morphogrammatic structure of transjunction re-occurs.
But let us, for argument's sake, assume that Rosser is right and we have to deal with 256 possible kinds of conjunction in a three-valued system. What shall we do with this embarrassing wealth? Rosser himself gives the answer: "Apparently the only thing that can be done about the matter is to pick out the 'and' that one likes best, and try to ignore the rest. " Emphasis by G. G.). McCulloch pointed out that the arbitrariness which Rosser suggested could not be tolerated in the development of a more basic theory of natural numbers. But he added meditatively: It hints at something in the relation between matter and form. The author is not quite clear whether this was McCulloch's exact wording; at any rate, he asked his mentor what he meant and McCulloch spun a long tale which seemed to the hearer to go far beyond what he had learned from the essay' "What is a Number that Man may know it ...?". Finally a spark of tentative understanding jumped from the speaker to the listener. McCulloch was talking about Hermeneutics and about the possibility that, if numbers were subject to hermeneutic procedures in the sense of Dilthey's 'Verstehen' in the Geisteswissenschaften, this would definitely close for the scientist the gap between Nature and Geist. The idea of a basic 'arithmetization' of the Geisteswissenschaften seemed to the author at that time not only bizarre but outrageous and he voiced his violent objections. McCulloch did not answer any of them; he only asked curtly: and what do you make of Rosser's "sidewise motion"? (The reader who is not familiar with this paper should be informed that Rosser said in his somewhat loose manner that the mapping of natural numbers on a many-valued logic produces something like a "sidewise motion" of these numbers.)
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