Goedel_GG_000.html

Gotthard Günther (1900-1984) began an academic career in Germany and published a book on Hegel's logic (Günthier 1933), based on his 1932 dissertation. He left Germany in 1937 and after brief stays in Italy and South Africa came to the United States in 1940. In the 1940s he taught for a time at Colby College, but he did not have a regular academic position again until 1961, at the end of his correspondence with Gödel, which begins in 1953. During that correspondence he lived in Richmond, Virginia, where his wife seems to have been employed. He had research grants and earned money as a flying instructor and by freelance writing. During the correspondence his relation with the University of Hamburg began, where he obtained a visiting position for the winter semester of 1955-1956. In 1961 he became a research professor in the department of electrical engineering at the University of Illinois in Urbana. He was given the title of Professor Emeritus at Hamburg, and in 1971, after his retirement from Illinois, he moved permanently to Hamburg (after further visits) and continued to give lectures there until the 1982-1983 academic year.

Günther's body of writing is considerable, but it is unlikely to be known to most readers of Gödel's works. His original philosophical background was Hegelian, and he continued to see philosophy from that point of view, though he was also influenced by Leibniz and by twentieth-century German figures. Moreover, although he lived for 30 years in the United States, even during that period his philosophical writing was mostly in German.[³] A project that he pursued for many years, which is one of the themes of his correspondence with Gödel, was how formal logic ought to be revised to accommodate what he took to be insights about the nature of thought and its relation to reality from the German idealist tradition. He also became interested in and wrote about "cybernetics". Norbert Wiener, who publicized the term, characterized cybernetics as the science of "control and communication, in the animal and the machine."[⁴] Its concerns derived from engineering and theoretical biology, but what seems to have most interested Günther was the idea of artificial intelligence. He was one of the earlier thinkers to write from a philosophical point of view on that subject.[⁵] He was thus a very unusual intellectual figure for his time, a Hegelian philosopher with an interest in modern logic and involvement in what later came to be called computer science.

The occasion for Günther´s correspondence with Gödel was an inquiry Gödel received from the American Committee for Emigré Scholars, Writers and Artists, which wished to support an application by Günther for a grant from the Bollingen Foundation. Günther followed this up by writing to Gödel on 2 August 1953 and sending him several papers. Shortly thereafter he took the liberty of using Gödel´s name as a reference. Gödel wrote what was evidently a supportive letter, of which handwritten drafts survive in his papers.[⁶] On 12 December 1953 Günther wrote to inform Gödel that he had received a three-year grant and to thank him for his support. The more substantive correspondence began the following spring. On 29 April 1954 Günther wrote expressing some views and raising some questions about the law of excluded middle. The exchange continued for several years, with Günther, however, writing more and longer letters than Gödel. On 17 September 1956 they met in Gödel's office and had a morning of discussion, apparently largely of Günther's work.[⁷] Gödel's last known letter to Günther was in January 1959, but Günther continued to write to Gödel through 1959 and 1960. There is no evidence known to us that G6del sent any further replies.

Before we describe the course of the correspondence, it is necessary to say something about Giinther's point of view at the time and about the philosophical project, aspects of which he lays out in his letters, in whose service he was making inquiries with Gödel. The conviction with which he began and which animated his whole involvement with logic is that the philosophical insights of German idealism from Kant through Hegel required a revision of logic. This view was already expressed in the published version of his dissertation.[⁸] Probably by the time he left Germany he was convinced that modern mathematical logic was relevant to the project of such a revision,[⁹] but he was never convinced that in its classical version it was what was required. He considered Hegel's logic an effort in this direction, although he was aware that it was not a logic in the sense in which the systems constructed in modern logic are logics. He thought Hegel's logic a grand failure, but he remained interested in the project of expressing it in more formal terms. That logic should be intimately related to metaphysics was a lifelong conviction of his and no doubt a point of agreement with Hegel. Moreover, to express his view of metaphysics he constantly uses as basic categories thought or consciousness, its relation to objects, and self-consciousness. In this respect the idealist tradition determines how he describes even non-idealist philosophy. Both the perspective from idealism and the metaphysical conception of logic are epitomized in his remark, "A logic is the metaphysical self-definition of a subject" (1957, p. 29).

Günther describes his project repeatedly as that of constructing a "non-Aristotelian logic." By "Aristotelian" he means what we would call classical, so that most of the vast extension of logic that has taken place since the mid-nineteenth century still counts as Aristotelian.[¹⁰] However, what is decisive for him is a certain metaphysical interpretation of the foundations of logic. His own preferred means for carrying out the project of a non-Aristotelian logic is many-valued logic. That is not itself non-Aristotelian, but it offers the technical means of carrying out the idea of a non- Aristotelian logic. Or so Günther thought in the 1950s at the time of his exchange with Gödel.[¹¹]

Günther's conception of classical logic is intimately bound up with a view of the relation of logic and metaphysics. He sees the metaphysical tradition from Plato and Aristotle at least through pre-Kantian modern philosophy as a unity and classical logic as obtaining its rationale and metaphysical foundation from that tradition. He describes as "axioms" of the classical tradition in logic the principles of identity, contradiction, and excluded middle (1957, p. 5). Whenever he discusses the foundations of logic, however, the framework is that of a theory of thought, in particular of a subject thinking about an object, or being conscious of an object.

What is most distinctive in Günther's point of view comes out in his remarks about the principle of identity. Repeatedly he says that the metaphysical tradition is based on the presupposition of the identity of thought and being. He also took this as a metaphysical presupposition of classical logic. As he said in his letter of 23 May 1954 to Gödel, "Classical logic presupposes the metaphysical identity of thought and being" (letter 3, p. 17). The "original phenomenon" of thought is expressed by "I think something." The relation of "I" and "something" is characterized as identity. What this comes to is that the object, the "something" is identical with itself, but also that the ego is in the end identical with its object.

While consciousness in the judgment "I think something" determines its definitive and final subject matter, that is the "something", simply as "identity", it claims implicitly that all thinking that is possible at all intends as its definitive metaphysical goal the objective In-itself, identical with itself. The transcendent essence of all self-identical In-itself, however, is Being. Consequently Being is the only, original, and last metaphysical subject matter of reflecting consciousness (1957, p. 8).

The object of thought can preserve its identity through the different perspectives from which it is experienced and thought about. Thus that it is identical with itself means not simply what is expressed by the logical truth that everything is identical with itself, but, one might say, that the concept of identity has a non-trivial application, so that the same object can be presented to consciousness under different circumstances and in different ways, which we might call modes of presentation.

A clear conclusion from the identity of thought and Being that Günther draws in his principal systematic work, Idee und Grundriss einer nicht-aristotelischen Logik,[¹²] is that what is objective or what is true will be perfectly intersubjective (1978a, p. 11). If a subject has a "true concept" of an object, that will in principle be communicable to any other subject. But Günther sees this as implying that the division of subjectivity in general into individual subjects is only "provisional and apparent." I think what he means by this is that an intrinsic goal of the thinking of individuals is to converge on some absolute thinking, in which each one's thoughts would represent the world as it is and thus not in any way differently from those of any other. This train of thought leads to the idea of an absolute subject, in effect God, in whom the relativity of consciousness to a perspective would be overcome. One of the challenges to this way of thinking is the thought, prominent in post-Hegelian philosophy, that actual subjectivity is formed by history (1978a, p. 10).

To get a fuller idea of Günther's conception of classical metaphysical thinking, we have to see a little of what he thought overthrew it. One development he mentions but does not give a prominent place in his philosophical analysis is the gradual divorce of fundamental science from philosophy. It is because modern logic arose primarily within mathematics, and thus on the scientific side of this divide, that Günther thought it did not break with the classical logical tradition on the metaphysical level.

What is central to Günther's story is German idealism beginning with Kant but especially as embodied in Fichte, Schelling and Hegel. His picture of philosophy before Kant clearly owes much to idealism; for example he views the classical tradition as in Kantian terms transcendental realist. However, if one thinks of pre-Kantian philosophy as transcendental or metaphysical realism, one will see the main revolutionary element in German idealism as Kant's Copernican turn and its further development by his successors. That is for Günther only part of the truth. What is central, though related, is the role self-consciousness plays in the thought of these philosophers. It is attention to subjectivity that was lacking in earlier thought. To Gödel he writes the following:

In the classical tradition the subject of thought and the process of reflection do not count at all. The goal of thought is to grasp the sense of absolutely objective being. And truth means absolute agreement of thought with the absolutely objective object. That is, all categories of logic must, if they are to be true, be absolutely objectively definable. Everything "subjective" is quite simply to be eliminated (letter 3, p. 3).

Now, whatever one might think of this as a characterization of ancient and medieval thought, it is a commonplace that subjectivity becomes a philosophical theme with Descartes. Günther has much more to say about what is new in Kant and his successors; the central concept that he uses to describe it is "reflection" - It is not easy to say what he means by it. A basic meaning is certainly "self-consciousness". More generally, it is represented as a feature of thought about objects when the conception of the objects takes into account the subject's thought about them. In this way it comes also to cover what I myself would call semantic reflection, that is, the passage from the straightforward use of words "taken at face value" to discourse in which they are mentioned and something is said about their reference, truth or meaning.

Some helpful explanations are found in Günther 1957 (already quoted above). In this paper Günther distinguishes stages of reflection, which are fundamental stances of consciousness to the world and itself, which, however, also have some relation to stages in the history of thought. The stages are described as "R-levels", levels of reflection. The idea seems to have some inspiration from the theory of types (1957, p. 5 n. 1), but also from Hegel's Phenomenology of Spirit.

The first R-level is, as one might expect, that at which the 0th level becomes an object. Günther describes it as arising from an "existential contradiction" in the 0th level, because at that level the subject realizes itself as negation, as mere object, that is as non-subject. But at the first level it is able to distinguish being and consciousness. It discovers that being is mirrored in consciousness, but nothing else is so reflected. So consciousness is in reality being.

A second R-level, which is conscious of the first level and its limitations, seems to arise at the end of the Middle Ages. Although he agrees with the historians of logic that nothing came of it for formal logic, he seems to attribute some awareness of the limitations of Aristotelian logic, which as belonging to the first level can only be about the 0th level, to those who rebelled against medieval logic during the Renaissance. But no new logical system was possible at the time.

Every R-level, according to Günther, can itself be object of a further reflection. Thus the iteration involved is infinite. Günther draws the further conclusion that it cannot at this point be characterized axiomatically, as a new logic would require, "because it is in no way possible to arrive at final, most general propositions about this open subject and to define it as self-consciousness (thus as a closed whole)" (1957, p. 23). Günther had stumbled on a kind of paradox, which is familiar from reflection on the theory of types. If "reflection", whatever that is, is iterated an arbitrary finite number of times, one arrives only at a particular stage of reflection from which it is possible to go a step further. Therefore general propositions about, all stages can't be formulated.

Günther formulates the problem as one about self-consciousness. At the second level consciousness is related to a relation of consciousness and objects, but self-consciousness is not part of what one is conscious of. It seems that at each level from the first on one has self-consciousness which is, however, limited to its own consciousness at lower levels and so does not really take in itself. Günther argues that this leads to a predicament in which metaphysics is impossible.

In the theory of types, eventually a way was found of so formulating the theory that the iteration of progression to higher types can be iterated into the transfinite, and of course with variables of type W one can make statements about entities of arbitrary - finite types. This does not abolish the dilemma, however, because the progression to higher types can be carried still further. The understanding of the language of set theory that comes most naturally takes the quantifiers as ranging over absolutely all sets, and thus as encompassing all types in the simple type hierarchy. That is not the end of the story, however, since that interpretation is not beyond question, and one still has to cope with another form of ascent, the semantic ascent that, in a perfectly classical setting, parallels ascent in Russell's ramified hierarchy.[¹⁴]

Günther sees the achievement of idealism as at least posing the task of developing a conception of self-consciousness that would have the required closure property, so that self-consciousness would be truly consciousness of itself and not just a representation of itself as an object. In developing his own scheme he does not stay with the idea that there will be an nth R-level for every n, and on into the transfinite. That there is a higher standpoint is, according to Günther, shown by the fact that it is possible to conceive such an iteration and make judgments about it, whether or not one holds that there is a higher standpoint.

With this thought of the infinite iterability of the reflected consciousness, we have already elevated ourselves above the infinite series of reflections proceeding from the second R-level and made it the "object" of a reflection that by definition cannot itself belong to this sequence. The content of this new reflection is thus the idea of the totality of the infinite sequence of iterations (and not itself an iteration on which others can follow) (1957, p. 27).

Günther concludes that we have a true third R-level that is not just a term in the sequence of iterations of reflection, in which the subject reflects on itself and thus "defines the ego as total self-reflection". It might seem that Günther denies Hume's famous point and thinks that one can directly capture oneself in one's thinking. If that were so, the elaborate story about levels of reflection would be unnecessary. On the contrary, following Hegel, Günther maintains that the compulsion of consciousness toward objectivation can itself be "reflected" and thus seen as a feature of consciousness. It is that that makes "total reflection" possible.

In other writings Günther uses a scheme that he finds in Hegel, which finds three different levels of reflection: "Reflexion-in-anderes", the simple thought of an object by a subject; "Reflexion-in-sich", the thought by a subject of a subject that, however, plays the role of object; and "Reflexion-in-sich der Reflexion-in-sich-und- anderes",[¹⁵] which seems at first sight to be just the thought by a subject of a subject's thinking of an object, where both the subject thought of and the object are objects of the first's subject's thought.[¹⁶] There is a rough correspondence between these and the 0th, 1st and 2nd R-levels, but Günther uses these concepts for different purposes, in particular explaining the truth-values of his three-valued logic, as we shall see shortly.

One of the other purposes is the "deduction" of the notion of the Thou (Du). Given the fundamental role of consciousness in his thinking, and its distinctness from "being", it is not surprising that other minds should be an ontological category in their own right. The Thou is a subject, thus with the same reflective closure we have been considering. That taking the Thou seriously should lead to a fundamental change follows from Günther's view of earlier philosophy, because the identity of thought and being tended in the end to abolish the differences between subjects.

This view leads, according to Günther, to a questioning of the law of excluded middle. In some way he identifies the rejection of a "third" between subject and object and the rejection of a third truth-value, or an alternative to being true and being false. The "home" of the law of excluded middle is a classical subject-object schema in which one or more subjects relate to an object. But once one accepts theorem I, two subjects will not be equivalent in the way the tradition had it. If two subjects think of an object, one will be able to think of the other as thinking of the object. That is what Günther calls "Reflexion-in-sich der Reflexion-in-sich und anderes", schematized as SS Æ (SO Æ OS). This suggests an asymmetry that is, however, not the final point of view because it leaves out the fact that the other subject is also capable of thinking of me. We have, in Husserl's well-known phrase, transcendental intersubjectivity.[¹⁷]

Günther argues that from this point of view one must distinguish two negations, one of which is expressed in the statement that the subject is not its object, another in the statement that the subject (as ego) is not the Thou. Günther seems to be driven toward many-valued logic by the fact that he doesn't consider an alternative to a truth- functional interpretation of propositional logic, and at least a third truth-value is needed in order to make the distinction between the two negations.

How, then, does he interpret his "truth"-values? At this point he does something that is from a logician's point of view crazy, because the values seem not to be truth-values at all. He uses the designations I, R and D, which he reads as "irreflexive", "reflexive" and "double-reflexive". In other words, they represent stages of reflection coming out of the analysis we have discussed. He even says in one place that all the values are "true" (1953, p. 48). The concepts of truth and falsity should "disappear without remainder" from the sort of logic he is constructing because they exclude a genuine third.[¹⁸] It seems that he has simply changed the subject, as a result of taking the relation of the I to what is not I as the paradigm of all negation.

Günther is, however, a somewhat slippier target, and I don't think I have grasped his thought at this point. He says that our thought is in a way necessarily two-valued. What the three-valued logic does is allow for the fact that two-valued thought can occur at different levels of reflection. How he conceives this is not at all clear to me. But he does say something about how it works in propositional logic. He singles out pairs of values and notes that one might treat that pair as truth and falsity, and certain functions might behave like, say, conjunction when just these two values are considered, perhaps behaving differently when the third value is taken into account. He saw the fact that two-valued structures can occur in different places in a three- or more-valued system as analogous to the place-value feature of Arabic or binary notation for numbers:

A many-valued logic is now nothing but a system that allows us to give to our single "actual" logic different place-values in the system of consciousness of such a kind that each place-value is connected with a different semantic meaning of the two-valued calculus that thus repeats itself. Such a many-valued system allows us thus to read off the structural interrelation of the different two-valued stages of consciousness.[¹⁹]

Günther wrote Gödel on 29 April 1954 with questions about the law of excluded middle, prompted by an allusion in Menger 1933 to Gödel's translations of classical logic and arithmetic into intuitionistic. Although Menger gives a reference to Gödel 1933e, Günther says he cannot obtain it and struggles with the issues as best he can. He concerns himself with the difference between a version of the law of excluded middle in propositional logic and the quantificational schema ÿ"xFx Æ $xÿFx, which he regards as a formulation of the law although it is better described as a consequence of it. His comments on the latter are not very clear, but he seems to hold that its truth in a given case presupposes that an object is given that satisfies ÿFx if anything does.

Gödel's reply of 15 May to this rather confused letter is a gem, an introductory lesson on the relation of classical to intuitionistic logic and mathematics. In the context of setting forth some of the basic technical facts, he agrees with Günther that the basis of the difference of intuitionist and classical logic is that they apply different conceptions of being. He adopts Günther's term 'aufweisbare Existenz' ('exhibitable existence') for the conception at work in intuitionism. "But the core of the intuitionistic objections lies surely in the fact that it is shown that for exhibitable existence certain statements of classical mathematics are unproved and others are even demonstrably false."

Gödel notes a point of philosophical agreement with Günther, that "the application of the method and results of mathematics [in particular, no doubt, mathematical logic] should not be limited to positivistic philosophy." Günther had remarked that he had been trying, thus far unsuccessfully, to persuade his fellow metaphysicians to pay attention to modern logic.

In his reply of 23 May 1954 Günther does more than before to set forth his basic ideas about the connection of metaphysics and logic. Some of the general ideas discussed above, in particular the connection Günther sees between classical logic and a certain metaphysical tradition, are sketched, and he introduces the idea of three-valued logic, which in his writings of this time is the main technical device for carrying out his logical program. Some of the discussion (following up the earlier exchange on intuitionism) is about different conceptions of being, and early in the letter Günther poses a challenge to Gödel. He remarks on the aim stated in Gödel 1944 "to set up a consistent theory of classes and concepts as objectively existing entities" (p. 152) and asks in terms of which of the different conceptions of being the "objectively existing entities" should be understood.

One might object to Günther's point of view in many ways. Gödel's response (letter 4, 30 June) is limited in what it takes on. He first denies, that Günther's philosophical claims contradict his own results "although my results make impossible certain forms of a subjectivistic interpretation of mathematics and in general speak strongly against every such interpretation." He remarks on the relevance of undecidability theorems for the law of excluded middle.

The reflection on the subject treated in idealistic philosophy (that is, your second theme of thought), the distinction of levels of reflection, etc., seem to me interesting and correct. I even consider it entirely possible that this is "the" way to the correct metaphysics. However, I cannot go along with the denial of the objective meaning of thought that is connected with it, [although] it is really quite independent of it. I do not believe that any Kantian or positivistic argument or the antinomies of set theory or quantum mechanics has proved that the concept of objective being (no matter whether for things or abstract entities) is senseless or contradictory. When I say that one can (or should) develop a theory of classes as objectively existing entities, I do indeed mean by that existence in the sense of ontological metaphysics (pp. 3-4).

Whether idealist philosophers deny "the objective meaning of thought" is a highly controversial matter. I don't think Gödel has really assimilated Günther's conception of "ontological metaphysics", which on his view is an expression of the identity of thought and being that he associates with the classical tradition. Gödel may be affirming a form of transcendental realism, but the very fact that he thinks it can coexist with what he agrees to be insights of idealism implies that it is not exactly ontological metaphysics as Günther understood it. The matter was pursued only to a limited degree in the correspondence. But it illustrates a puzzle about Gödel's philosophy that is not confined to this exchange. Gödel is reported by Hao Wang to have much later described his philosophy as idealistic (Wang 1996, remark 0.2.2, p. 8), and after 1959 he was much attracted to the thought of Husserl, who described his position as transcendental idealism. Yet Gödel, in writing about mathematics and in a few places (including the above) about physics, expresses strong realistic convictions. He gives little explanation of how realism and idealism can coexist.[²⁰]

Günther wrote two letters (2 October 1954 and 19 June 1955) before Gödel replied again. The issues he raises again concern the sense in which objects of different kinds are independent of thought. He mentions again some realistic remarks in Gödel 1944 and quite reasonably queries the sense in which objects are said to be independent of thought (or of something else belonging to "us", e.g. "our definitions and constructions" (Gödel 1944, p. 134)). What concerns him is that different objects might be independent in different senses, although according to the first letter "the previous theory of thought from Aristotle to the present knows only one concept of logical (thought-independent) object!" This could be questioned, but it might be more relevant to object that classical formal logic does not distinguish different concepts of object or of thought-independence simply because the relevant differences are not in the province of formal logic; that need not make logic inapplicable across these differences. Perhaps intuitionistic logic does in its intended application call for a different conception of object, but it does not follow that classical logic is not applicable to a number of different conceptions.

It's not surprising that in the next letter Günther gives some exposition of three-valued logic. But the context is a different line of thought, the distinction between reflection on objects thought of as independent of thought and reflection on reflection itself. He claims that these two types of reflection require fundamentally different concepts of object and the difference consists in whether the object is altered by the act of thought directed at it. He maintains that this obtains when one thinks about one's own subjective act of thought.

Gödel finally replied on 10 August 1955 (letter 7). He directs his criticism mainly at the formulation in Günther's second letter; even in intuitionistic logic, which he concedes is "a reflection on thought", objects of thought are as little altered by acts of thought directed at them as the objects of physics. He agrees, however, that objects of the second type of reflection are totally different from objects of the first. This letter also contains Gödel's most extended comment on Günther's ideas about many-valued logic; see §3 below.

There followed a series of letters of Günther to which we have no replies by Gödel. Only the first (letter 8, 18 September 1955) represents an attempt to respond substantively to Gödel. On 22 June 1956, after returning to Richmond from his visit to Hamburg and a few months' stay in Chicago,[²¹] Günther asked for Gödel's support in his application for a renewal of his grant from the Bollingen Foundation and sent him a manuscript which appears to have been a preprint of Günther 1958, his most extended exposition of the ideas on many-valued logic in relation to metaphysics. On 17 September they met for a morning in Gödel's office, and evidently they discussed this paper.[²²] Gödel wrote to the Foundation on 20 October to express his "whole-hearted support.[²³] He says that the recent paper "Die aristotelische Logik des Seins und die nicht-aristotelische Logik der Reflexion" (i.e., Günther 1958) "fully confirms my favorable judgment".[²⁴] Gödel's final remark is revealing about what attracted him in Günther's project:

On 26 February 1957, when Günther was in California for surgery, he wrote developing the idea that the concept of reflection really originates with Leibniz, although as a logical term "reflection" derives from Hegel. He then goes on to develop some interpretation of Leibniz, using some of his own conceptual apparatus. When Gödel next wrote, on 4 April 1957 (letter 10), he reacted skeptically, finding the ideas interesting but doubting their basis in Leibniz's text. Following up their discussion in Princeton in 1956 of the manuscript of Günther 1958, he says that its basic ideas need clearer explanation and elucidation by examples. In its present form it is "hardly intelligible". But Gödel expresses interest in an idea Günther had expressed earlier, probably in the manuscript that he sent in 1953, that "total reflection" would be something going beyond all ascent of types. (We will return to this subject in §3.) This pleased Günther enough to overcome the effect of the critical tone of the rest, as his reply of 7 April shows. He then sent Gödel some of his papers. Gödel drafted a short note, dated 2 May, acknowledging their receipt.[²⁷]

It seems likely that Gödel's interest in Günther's ideas had begun to decline. In the remainder of 1957, Günther wrote only one substantive letter (22 November, with copies of Günther 1957 and 1957a). Gödel replied on 23 December (letter 13) with rather brief comments on some of the papers Günther had sent. About Günther 1957, he says it has some overlap with a manuscript Günther had sent him previously, probably the one sent in 1953.[²⁸] He says he has read some of "your new work", apparently 1957a[²⁹a] and praises its clarity, particularly in contrast to the version of 1958 that he had read.

During 1958 Günther wrote Gödel twice, sending new papers. Gödel did not reply; when Günther wrote on 1 January 1959 expressing concern, Gödel wrote an interesting reply (letter 14, 7 January), again referring to Günther's earlier idea. But that was evidently his last reply. Günther continued to write and send papers through 1959 and 1960 but got no response from Gödel. In 1961 Günther obtained the appointment at the University of Illinois. The last item in the correspondence is an undated notice of the change of his address from Richmond to Urbana, postmarked 27 July 1961. On it Günther wrote, "I have accepted a research professorship at the University of Illinois and I am looking forward to it. Herzlichst, Ihr G.G." He was perhaps letting Gödel know that he too no longer needed to continue the exchange.

Regarding the meaning of the truth-values, Günther had attempted an explanation in his letters and in at least one publication before that time, Giinther 1953. But in the same letter Gödel wrote that the manner in which Günther wanted to introduce the third truth-value was "not wholly intelligible" to him and that he missed any explanation of what the three truth-values really mean. Gödel may not have seen enough by August 1955 to have even the limited grasp of Günther's intention expressed in §1. Probably he had by the time he read the manuscript of Günther 1958, but this did not lead to a change in his expressed attitude, and his view of that paper was not very favorable. It is easy to guess what the obstacle was: Günther wasn't able to give a motivation for modifying basic logic independent of his particular metaphysical interpretation of classical logic, and although he explains the three values in terms of his ideas about consciousness, he not only does not make clear how they are truth-values but seems to want to cut loose from the concept of truth, more so even than intuitionistic logic does. One might well ask whether what resulted should still be called logic.

Günther also seems never to have tried to write down axioms and rules of inference for his many-valued calculi. Probably that would have been necessary to persuade Gödel that this was intelligible logical work. But one can see why not only technical limitations prevented Günther from undertaking the task. If the "truth" values express levels of reflection, what would be the significance of being a theorem of the calculus? Such calculi are formulated in order to characterize conceptions of valid logical inference or logical truth. It's not clear that Günther's scheme has any place for these notions. That the concept of truth is not at center stage for him is indicated also by the fact that when he writes about intuitionism, although it is clear to him that in some sense the conception of existence is different from that in classical mathematics, he never remarks on the fact that there is a more underlying difference about truth. Gödel, however, does not make a remark that would smoke Günther out on this issue.

I have left for the last what is probably the most interesting of Gödel's reactions, on the matter of the relation of levels of reflection and type theories. As I have said, it is something Günther said about this that seems to have piqued Gödel's interest at the outset and made him tolerant of Günther's evident technical limitations. It is a theme that Gödel comes back to a couple of times without being prompted by Günther.

We have to reconstruct what Günther's idea may have been, since I have not been able to locate the manuscript Günther sent in 1953. Gödel found some overlap between that manuscript and Günther 1957, and as we saw above, Günther does formulate an idea of "total reflection" and claim that the third R-level at which that is achieved does contain within itself all iterations of reflection and so the hierarchy of types. In Günther 1957 no connection with a type-free logic is made, but it seems that in the manuscript of 1953 such a connection was made. It doesn't appear, however, that any concrete suggestions for such a logic were offered, and Gödel had no success in prodding Günther to work out the idea, even though it was pressing this idea that most pleased Günther in Gödel's letter of 4 April 1957, as he wrote on 7 April. Very likely Günther's invocation of many-valued logic at this point disappointed Gödel.

Evidently Gödel entertained hopes that the kind of analysis of the structure of self-consciousness that Günther undertakes and finds precedent for in the classical idealists would yield not only philosophical illumination but even a basis for constructing a type-free logic that would be philosophically better motivated than what was then available. In his last letter, letter 14, he even says that the correct axioms for such a logic should follow "with necessity from philosophical insights about the essence of reflection." Much earlier he had made remarks that suggest that the most satisfactory theory of what he calls "concepts" would be a type-free logic of some kind.[³¹c] It is not known that Gödel did any substantial work toward the construction of such a logic. If he had something in hand early in his exchange with Günther, it is imaginable that he would have brought it up. But it seems to me more likely that he was very uncertain of how one might proceed and therefore perhaps disposed to entertain hopes that light might come from somewhat unlikely sources. Gödel seems never to have resolved this uncertainty, as is indicated by his remarks in his conversations with Wang about the theory of concepts and the unsolved nature of the "intensional paradoxes".[³²d]

What does this exchange tell us about Gödel's thought during this period, beyond what one can learn from remarks he makes in his letters? What he did not say is revealing. He nowhere says that in order to attain a philosophical understanding of thought and its relation to objects, one might well begin with logic rather than taking "subject" and "object" or "thought" and "being" as fundamental categories. Such an outlook might, indeed, lead us to hold out greater hopes for understanding self-consciousness with the help of the logical analysis of semantic reflection, as it is carried out in theories of truth, than for the reverse procedure of expecting to motivate the axioms of a logical theory from an analysis of self-consciousness.

Still less does Gödel invoke the linguistic turn, for example what Michael Dummett regards as an axiom of analytical philosophy, "first, that a philosophical account of thought can be attained through a philosophical account of language, and, secondly, that a comprehensive account can only be so attained."[³³e] One shouldn't read too much into the absence of something from correspondence, but Günther's ideas seem so obviously to call forth a response of some such kind that we might discern at least an indication that the linguistic turn, at least as Dummett understands it, was not central to Gödel's practice as a philosopher.[³⁴f]

Given Günther's obvious technical limitations and the difficulty Gödel had in making sense of his proposals, we might find it surprising that Gödel maintained his interest in Günther for as long as he did and even up to early 1957 wrote supportive letters for him (see §2 above). There seems to have been some mutual personal sympathy; for example both write about their health problems and respond sympathetically to the other's. Both were outsiders to the American philosophical world and shared the project of dissociating modern logic from positivistic philosophy. But a deeper reason is very likely that Gödel was at the time occupied with philosophical problems about concepts (it was the time during which he worked on *1953/9) and was very unsure about how to approach them.[³⁵g] He was evidently prepared to entertain the possibility that post-Kantian idealism, to which he had apparently not had a lot of exposure, would be a source of illumination. He found Günther a clear expositor of ideas from that tradition.[³⁶h] But he does not seem to have been disposed to work out himself a line of thought in which self-consciousness is a central concept, and when Günther did not pursue what Gödel thought the most promising direction, he lost interest. Not long after his last letter he began his study of Husserl, whose version of idealism he seems to have found much more satisfactory.

A complete calendar of the correspondence with Günther appears on pp. 563-564 of this volume. The editors are indebted to Mr. Lothar Busch for locating Gödel's letters in Günther's papers while they were being catalogued, to Professor Tilo Brandis, Director of the Handschriftenabteilung of the Staatsbibliothek zu Berlin-Preussischer Kulturbesitz, for making copies of these letters available early in our work, as well as to Delia Graff and Øystein Linnebo for preparing typescripts of handwritten letters. The translation is by Thomas Teufel and Charles Parsons, revised using suggestions of John W. Dawson, Jr.

² Charles Parsons was educated at Harvard, receiving his A.B. in mathematics in 1954 and his Ph.D. in philosophy in 1961. After teaching briefly at Cornell and Harvard, he joined the faculty at Columbia University in 1965 and remained there until 1989, serving for most of that time as an editor of the Journal of Philosophy and for two terms as department chair. In 1989, he came to Harvard; in 1991, he became Edgar Pierce Professor.

² Professor Parsons has published papers on mathematical logic, philosophy of mathematics, philosophy of logic and language, Kant, and historical figures in the foundations of mathematics, such as Frege, Hilbert, and Gödel. Some of his philosophical papers are collected in Mathematics in Philosophy (Cornell, 1983). He is editor, with Solomon Feferman and others, of Volume III of the Collected Works of Kurt Gödel, Unpublished Essays and Lectures (Oxford, 1995), and of the two forthcoming volumes of correspondence.

⁵ Günther 1952 is his most direct discussion of whether machines can be conscious. It seems to be presupposed in Günther 1957a, the clearest and most accessible presentation of the ideas Günther discussed with Gödel, as GödeI himself seems to have thought (letter 13, but see note aa below). Since Günther 1963 incorporates the latter and contains a reprint of the former, it is the best introduction to his ideas.

⁹ See Günther 1940 and 1957, of which the latter appears to have been drafted in 1935; see Günther 1980, p. 305 n. Unlike other publications of Günther in the 1950s dealing with logic, it makes no mention of three-valued logic. Günther sent Gödel the published version without having mentioned the paper earlier in the correspondence; that would suggest he was not working on it during their exchange.

¹⁰ "We can leave out of account the 'logistic' criticism [of Aristotelian logic] beginning with Leibniz, because it represents no philosophical critique of the metaphysics of this logic but a generalization and extension (functional calculus) of the classical ways of proceeding" (1957, p. 5 n.). He does go on to say that his own work would not be possible without modern logic, and already in 1940 he argued that philosophers interested in "transcendental logic" needed to pay attention to modern logic.

¹¹ Günther 1958 sets forth what was to be the basic logical construction for the second volume of the work of which 1959 was the first. He states (1978a, p. xxii) that the calculus was not able to bear the philosophical weight it was meant to carry. He apparently came to this conclusion through exchanges with cyberneticists early in his time at the University of Illinois. No attempt is made here to follow Günther's thought after his exchange with Gödel.

¹² Günther 1959. This work was Günther's principal project during his exchange with Gödel, and its Preface contains a generous acknowledgment to him (1978a, p. xxi). Quotations from this work are from the second edition, Günther 1978a. Translations of Günther's published writings are my own, of his letters to Gödel by Thomas Teufel and me.

¹⁴ If, as I have suggested, "reflection" as Günther understands it includes semantic reflection, then complete closure, that is a theory that would express its own semantics without any remainder, is impossible. "Total reflection" is not so clearly defined that one can say definitely whether it implies semantic closure in this sense. An affirmative answer is suggested in a passage in Günther 1957a (1963, pp. 78-79). It is somewhat difficult to interpret because it is not always clear when he is rendering Hegel and when speaking for himself. Although the I/R/D three-valued logic in a way renders total reflection, it describes "a thinking that is not and cannot be thought by anyone" (p. 79). To capture the subject of this thought, one must ascend to a four-valued logic, and indefinite further ascent can be forced. Thus Günther writes in letter 11 that "total reflection is not three-valued but indefinitely n-valued, where n > 2 always holds". In the end he seems not to claim to escape an ascent like the ascent of types.

²⁰ Of course this is not to say that they cannot. In particular, Husserl has been interpreted so that his position is at least compatible with "common-sense realism", and many philosophers, including perhaps Husserl, have undertaken to overcome the opposition between realism and idealism or some other opposite of realism. In his introductory note to *1961/? (these Works, vol. III, pp. 364-373), Dagfinn Føllesdal evidently does not see an opposition between Husserl's "idealism" and Gödel's realism concerning mathematics.

²² See above, in particular note e. In a card sent from Hamburg on 20 December 1955, Günther had said he would return to the United States in April or May and would like to drop in on Gödel at his office in Princeton. No other correspondence discusses a meeting; the actual arrangement could well have been made by telephone. That they discussed a version of Günther 1958 is shown by Gödel's letter to Günther of 4 April 1957.

²³ A typed draft of this letter, with corrections in Gödel's hand, is in Gödel's papers (document no. 010767). Quotations are from this draft, whose text is in all probability very close to that of the letter as sent. That 20 October was the date of the actual letter is confirmed by the Foundation's acknowledgment (Nancy Russ to Gödel, 8 November 1956).

²⁴ This brief remark replaces the crossed-out statement that the paper "gives a remarkably clear exposition of Dr. Günther's leading ideas within the field of logic and also develops new means for building a system of formal logic on their basis." Gödel wrote much less favorably about the paper to Günther on 4 April 1957. John Dawson has remarked to me that Gödel avoided writing unfavorable letters of recommendation. For that reason what he says in such letters cannot always be taken at face value.

²⁶ Gödel to Helmut Schelsky, 8 January 1957, marked "abgeschickt". In his letter of 15 January 1957, Günther acknowledges receipt of a letter from Gödel dated 10 January, now lost. He immediately thanks Gödel for the recommendation and makes no other comment. Gödel's letter may have done no more than inform Günther that the recommendation had been sent.

²⁸ Gödel does not comment now or later on the issues about minds and machines that are discussed in Giinther 1952 and 1957a and are mentioned in Günther's letter, although he had already written on the subject in the Gibbs Lecture *1951. (Günther had included 1952 in the packet of writings he sent in April; see his letter of 7 April.)

²⁹a It is not clear whether what is being praised for clarity is Günther 1957a or 1957. The fact that Gödel does not more explicitly indicate that he is talking about the same writing as in the previous sentence speaks for 1957a, as does the contrast with 1958, since the latter two writings set forth Günther's ideas on three-valued logic, while 1957 does not.

³⁰b Such an idea has been explored by a number of later writers on semantical paradoxes, and it was also observed that in some three-valued logical settings universal comprehension can be preserved. For a survey with its own development of these ideas see Feferman 1984. In these developments truth and falsity have their usual meanings and the third value has the sense "undefined". That is evidently quite different from what Günther had in mind, as he seems to say explicitly in a parenthetical remark in letter 8, p. 2.

³⁷i I am indebted to Klaus Oehler, Claus Baldus and Markus Stepanians for information about Günther, to Øystein Linnebo and Thomas Teufel for assistance in locating writings of Günther and for discussion of his ideas, and to John Dawson, Solomon Feferman, Warren Goldfarb, Sally Sedgwick and Wilfried Sieg for helpful comments.