## Number and Logos

Unforgettable Hours with Warren St. McCulloch

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.)

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 þ 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 pap