husserl and hilbert on completeness - Springer Link

ULRICH MAJER

HUSSERL AND HILBERT ON COMPLETENESS

A Neglected Chapter in Early Twentieth Century Foundations of Mathematics

1. INTRODUCTION There are two main dangers in dealing with Husserl: The first is to read and judge Husserl through the glasses of analytical philosophy and in particular through those of Frege and his successors.1 The second danger is to understand and interpret Husserl exclusively from an internal or, as we call it in German, from an immanent point of view, as believing “Husserlians” usually do. I don’t know which mistake is worse, but I find the second approach much more boring, whereas the mistakes and distortions of the analytical perspective can be at least amusing and sometimes are in fact quite interesting. I hope I can avoid both mistakes by focusing primarily on what Husserl did as a mathematician in his book Philosophy of Arithmetic [PA], and not so much on what he himself or others said, he did. Now, let me explain a bit more closely, what I intend to do. I will focus your attention on a particular piece of Husserl’s work, which has been dreadfully neglected.2 What I have in mind is Husserl’s so called ‘Doppelvortrag’, a pair of lectures that he presented before the Mathematical Society in Göttingen in the winterterm 1901, shortly after he had become “Extraordinarius” in Göttingen.3 Now, why is this Doppelvortrag so important? What precisely makes it so extraordinarily interesting? This simple question has several, distinctive answers, or better, it has one answer with several aspects, that all contribute to the significance of the Doppelvortrag [DV]. The principal answer is this: In these lectures Husserl inquires a question or problem, which is of tremendous significance not only for the foundations of arithmetic but for our logical apprehension of a deductive theory in general. The question is basically this: Under what conditions does the truth of a theory follow from its consistency? In order to grasp the significance of this question, one has to make clear its precise sense. I will do that later. But to convince you of its Synthese 110: 37–56, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

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exceptional importance, it’s sufficient to remark that Hilbert had expressed a similar view in a letter to Frege 1899, namely that the truth of a theory follows from its consistency and not vice versa. Whether that is correct or not, I will not discuss now. However it shows that other mathematicians had a similar vision, as it underlies Husserl’s question. And indeed, if it could be proved that the truth of a theory follows from its consistency the search for an indubitable foundation of mathematics would be a tremendous step forward: All that would remain to do is to prove the consistency of arithmetic in an absolute sense, i.e. without presupposing the truth of any theory whatsoever. But, as you know, these things are not as easy. And Husserl was quite aware of this, as will become clear, if we go on. But first let me mention two other reasons for the significance of the Doppelvortrag. Husserl himself stressed repeatedly the crucial role of the Doppelvortrag for the further development of his thoughts.4 Although, one has to be very careful in general with Husserl’s judgements about his own, earlier works, (because he often played down its true significance in order to improve the glory of his latest achievements), I think that in this case his judgement is basically correct. The reason for my estimation is quite simple and straightforward: Once you have really understood, what Husserl is up to in his PA, you will recognize that this book is in a certain way unfinished, that it entails a problem, to which Husserl had no fitting answer, at least not within the scope of that book. However, ten years later, in his DV he seemed to be very close to a “solution” of the old, unsolved problem, at least as far as we know, because we posses only a draft of the Doppelvortrag. There is a third more remote reason, why Husserl’s DV before the Mathematical Society in Göttingen is so important. This has to do with Husserl’s relation to D. Hilbert, his new colleague and promoter in Göttingen. Already the schedule of the Mathematical Society (as we find it reprinted in the “Jahresbericht der Deutschen Mathematiker-Vereinigung”) is instructive: II. Sitzung am 5. November [1901] ¨ D. Hilbert: Uber die Grundlagen der Geometrie und der Arithmetik. Der Vortragende bespricht zunächst das in dem Jahresbericht der Deutschen Mathematiker-Vereinigung Bd. VIII (1899) aufgestellte System von Axiomen der Arithmetik und hebt besonders die Bedeutung der Zweiteilung des Stetigkeitsprincips in das Archimedische Axiom und das Vollständigkeitsaxiom hervor. Weiterhin zeigt der Vortragende, wie sich die Lobatscheffskijsche Geometrie in der Ebene, ohne Zuhilfenahme räumlicher Axiome und ohne die Voraussetzung der Stetigkeit, auf Grund der Axiome der Verknüpfung, der Anordnung und der Kongruenz aufbauen läßt.

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V. Sitzung am 26. November [1901] D. Hilbert legt das 3. Heft des 55. Bandes der Math. Ann. vor und bespricht u. a.: M. Dehn: ¨ “Uber den Rauminhalt”. E. Husserl: “Der Durchgang durch das Unmögliche und die Vollständigkeit eines Axiomensystems”. – Daran anschliessend Diskussion – VII. Sitzung am 10. December [1901] E. Husserl setzt seinen Vortrag vom 26. Nov. fort. Vor allem werden die Begriffe des “definiten” und des “absolut definiten” Systems auseinandergesetzt. Bei definiten Systemen und nur bei diesen ist der Durchgang durch das Unmögliche gestattet.

As you see, not only do we know the exact dates of Husserl’s lectures,5 but we can also gather with certainty that Hilbert was present at Husserl’s first lecture, because he gave at the very same meeting a report of Dehn’s work. But it becomes even more exciting, if we look to the themes, which Husserl and Hilbert talked about. Both dealt primarily with the notion of completeness. In Husserl’s case this is obvious from the title; in Hilbert’s case we have to take the text after the title into account. Furthermore, there was a lively and genuine discussion after Husserl’s talk, because the discussion is explicitly mentioned, which is quite unusual. But not only this, we even know that Hilbert raised an objection to Husserl’s talk, which I will discuss at the end of this paper. This objection has been simply overlooked until today, because it had been misplaced as part of Hilbert’s lecture, that he had presented a month before, and of which Husserl had taken notes.6 Now, all this is very interesting for a number of reasons: First, as most historians of mathematics know, Hilbert had not only invented the ¨ axiom of completeness [in his lecture “Uber den Zahlbegriff” in 1899], but this axiom had also troubled him for at least three years, because its logical status in geometry was not quite clear, to say the least. Second, it is most probable, indeed almost certain that Husserl chose his topic quite ¨ intentionally. Not only did he know Hilbert’s essay “Uber den Zahlbegriff” – it had been published in 19007 – but Husserl had also attended Hilbert’s lecture the month before and had taken notes of it.8 Therefore, he knew very well that Hilbert attributed great emphasis to the axiom of completeness. But there is more to this story, to which I will come next. As already mentioned there is a correspondence between Hilbert and Frege about Hilbert’s book “Grundlagen der Geometrie” and in one of these letters Hilbert rejects Frege’s assertion as unwarranted that “from the truth of the axioms [of geometry] it follows that they do not contradict each other”, and instead maintains the reverse, namely, I quote, “if the arbitrarily given axioms do not contradict one another [with all their consequences], then they are true, then the things defined by the axioms

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exist. That is for me the criterion of truth and existence”. We know that this assertion, this exchange of letters, was known to Husserl, because Husserl had made excerpts from the correspondence, which entail just this part of the dispute. But not only this. Husserl also added the remark: “Frege does not understand the sense (Sinn) of Hilbert’s ‘axiomatic’ foundation of geometry.” But this isn’t the point that I want to talk about.9 Instead I will draw your attention to another aspect. Despite his critical remark about Frege, Husserl must have been puzzled by Hilbert’s objection to Frege for two reasons: First, without any further qualification Hilbert’s assertion that the truth of a theory follows from its consistency is at least unintelligible, if not plainly wrong, because we can think of many consistent theories, that are by no means true in any plausible sense of the word true. Consequently, some explanation or condition is missing in Hilbert’s claim. But even if this is added in a logical appropriate way Husserl would still be puzzled, because in his genetic view of theories arithmetic is built up step by step, such that the fundamental evidence for the theory is never put in question. Therefore, it seems, a proof of the truth of arithmetic from its consistency must be quite pointless from Husserl’s perspective as it was from Frege’s. But not quite so! Somehow Husserl felt himself challenged by Hilbert’s objection to Frege and he tried to figure out under which conditions [from his own, genetic point of view] a claim like Hilbert’s could and in fact would make sense. This is at least my tentative understanding of the DV and we have now to inquire whether this makes sense, and in which way it sheds a new and interesting light on Husserl’s philosophy of arithmetic.

2. THE UNSOLVED PROBLEM OF THE PA In order to understand, why Husserl was so receptive to Hilbert’s objection against Frege, we must clarify in which sense his Philosophy of Arithmetic was an unfinished book. To this end I must give an outline of the plan of the entire book. I don’t know of any better and shorter way to achieve this than to quote Hermann Weyl’s summary of Husserl’s PA, as he presented it in his Habilitations-Vortrag in Göttingen in 1910.10 Here is Weyl’s summary: The concept of set and number has run through various stages during the development of the human intellect. At the first stage it concerns the proper idea of a totality [Inbegriff], which emerges when representations of several individual objects, recognized as such, are raised by a unified interest from the content of our consciousness and held together. On this level, the lowest numerals, say 2, 3, and 4, designate immediately recognizable differences of the mental act coming into function in forming the idea of a totality.11


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Let me interrupt the quotation for a second in order to point out that this is such a wonderful compact, precise and balanced description of the first part of Husserl’s PA that I know of nothing comparable. I will come back to it in a moment. For now let me continue with the summary of the second part: At the second level symbolic representations stand for the proper ones. The most significant result of this second period is the procedure of symbolic counting, familiar to every child, that permits us to differentiate according to their number sets containing more things. In this development a certain sense of possibility also plays an important role in that, in order to do justice to the external world, we do not feel tied to the contingent restrictions and shortcomings of our sense-organs and mental abilities.12

After having explained and defended[!] Cantor’s original introduction of the transfinite ordinals as concordant with the symbolic procedure of counting, Weyl adds the illuminating remark to possible critics of the phenomenological approach: A proper representation of infinite sets cannot be carried out in the sense that its elements cannot simultaneously be present in our consciousness as individually recognized content. This is also true in the case of finite sets consisting of a larger number of elements. But from these facts there can arise no objection as to the logical permissibility of these sets. Only in this sense of the impossibility of a proper representation of an infinite set is it correct to say: there is no actual infinite.13

After this interruption Weyl comes eventually to the third and last level of Husserl’s phenomenological approach to mathematics: Because we are compelled by other irrefutable reasons to introduce infinite sets – indeed analysis alone forces this – when finally we come to the third level, where we erect the theory of finite and infinite sets and numbers in a scientifically systematic way by setting up appropriate axioms, definitions and the consequences drawn from them. Given what was said previously, we will have no scruples with respect to this construction in replacing the system of concepts intended by another that is completely isomorphic to the first.14

Now, why is Husserl’s PA “unfinished’? Weyl’s description looks so perfect that it is hard to imagine that the PA would entail a serious gap in the construction of arithmetic. But Weyl is a very distinguished and polite man; he covered the gap in Husserl’s PA just by jumping over the unsolved problem and thus giving his audience the illusion they could follow him on the same safe road. But that is simply not the case, and I have now to explain “why”. The main reason, to put it bluntly, has to do with the step from the first to the second level, from the proper representation of a multitude of single objects to the symbolic representation of, let’s say, the integers. Why is this step so difficult to take? Nothing seems easier than that! We do it all the time since our early childhood! But you have to consider, in order to

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understand the real difficulty, that Husserl permits on the first level only very restricted means to recognize numbers – means, which in principle do not suffice to generate the number “1”, let alone the number “0” or negative numbers. And this is by no means an accident but the purpose of the entire phenomenological investigation. Husserl wanted to clarify the minimal means by which we actually form the “first” number expressions. As such means he identified two activities: (i) the “colligation” of individual objects – whether mental or physical does not matter – and (ii) the “intentional abstraction” to take the individual objects simply as “elements”, which we hold together by a kind of reflective attention. The products of these two activities are called “kollektive Verbindung” (collective connection) and “Vielheit” (plurality). It is important to remark that – despite the thousands of words Husserl needs to explain his genetic approach – these two concepts are the single ideas by which we form number terms, because number statements are nothing but answers to the question: How many elements belong to a certain plurality? Whether one likes this analysis or not, doesn’t matter in the moment. However, it explains, why the “first” numbers are the numbers 2, 3, 4 etc.15 The reason is simply that every number statement presupposes a “plurality” of elements. To reply “1” or “0” would just be the wrong kind of answer to the question “How many chairs are in this room?”, because both responses deny that there exists a plurality. The situation here is quite similar to Russell’s example in “On Denoting”. If somebody asks “Is the present king of France bold?” you will not answer Yes or No, but reply: “Wait a moment, something is wrong with your question; there is no present king of France”. Quite similarly, if someone asks “How many chairs are in this room?” he presupposes that there is a plurality of elements. Consequently one cannot answer “1” or “0” but only “2, 3, or more”. To reply “1” or “0” means, properly speaking, to correct the question in the same way as before and to say “Wait a moment, there is no plurality of elements”. Once more, whether you agree to this kind of phenomenological analysis or not doesn’t matter, but it explains obviously, why there is a difficulty in going beyond the “Urdomain” of numbers, as I will call the domain of immediately accessible numbers [in distinction to the so called natural numbers, which are not so natural after all].16 However, to acknowledge that there is a difficulty in the step from the first to the second level, from the Urdomain to the integers, does not mean that the difficulty cannot be overcome. On the contrary, one can abandon the too narrow restrictions (to the concepts of collective connection and plurality), and instead use other, more “liberal” means. This is precisely, what Husserl does in the second part of the PA, and what Weyl characterizes


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so tactfully as the “symbolic” representation of numbers at the second level in contrast to their proper apprehension in the first place. But the suspension of the restrictions as such does in no way guarantee that no further difficulties occur. On the contrary, we now face the inverse problem: Do we really understand the extended means of “symbolic construction”, or is there an unbridgeable epistemological gap between the first and second, between the proper and symbolic apprehension of numbers? The fear that there might be a gap between the first and second part of his PA was a problem that troubled Husserl over ten years [from 1891 to 1901], until he eventually convinced himself that he was able to close the gap by introducing the notion of a “definitive manifold” [definite Mannigfaltigkeit]. In order to understand the solution I have first to explain the epistemological nature of the gap more closely. To this end consider again the question: How do we go beyond the urdomain of immediately given numbers 2, 3, 4 etc? The answer Husserl gives in PA is very simple and straightforward but at the same time also rather unsatisfactory, at least from an epistemological point of view. The answer, much abbreviated, runs like this: In real mathematics we can “calculate”, that means we are able to perform certain operations with numbers like addition and subtraction, multiplication and division, exponentiation and root extraction without any restriction in principle. But if we try to do the same thing in the Urdomain we run into many restrictions, because the Urdomain is too small to perform a subtraction or division unlimitedly. Consequently, we have to expand the Urdomain by a set of “new” elements, which enable us to perform the operation, let us say, of subtraction or division unlimited. But these new elements are not numbers in the phenomenological sense, that is we cannot apprehend them as the immediate result of a collection of intentional present objects, bound together by our reflective attention, but have to take them as the intermediate result of a numerical operation, which steps out into the imaginary, that is into a domain of objects, that exist only in our imagination. In other words the so called “negative numbers” or “fractions” are nothing but symbols for entities, that we imagine in order to perform certain numerical operations. Their existence is, so to speak, relative to certain numerical operations. But this in turn means that there is a gap between our understanding of the Urdomain of intentionally present numbers and the expanded domain of imaginary elements like the negative or rational and irrational numbers. Hence, the difficult question arises: Can we bridge that gap? At first glance, nothing seems easier than this; we just “define” the new elements by the numerical operations in question. But things are not as easy as they appear! First, Husserl believed he had solved

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the problem, but later, after having finished PA, the doubts returned that the gap still existed. Because I think, he was right, let me briefly explain why the gap cannot be bridged by sheer definition. To make a long story short; the definition required to bridge the gap, has to be creative. But this is from a logical point of view a dangerous idea; in this regard Frege was doubtlessly right. Take for example the subtraction of two numbers, a and b, and ask: How is it defined? The answer has two parts:17

a > b, then there is a number c in the Urdomain such that b + c = a; so far no problem arises. But what happens, if a < b? In this case we must say: (ii) If a is less or equal to b the result of the operation is undefined. (i)

If

In modern terms, subtraction is only a “partial function” with respect to the Urdomain.

In order to define it completely for every possible pair of natural numbers, we have to create new elements ni and to add them to the Urdomain such that the general existence sentence 8a8b9n(b + n = a) becomes true. But now, you see, the question of the domain of the variables becomes crucial: (i) Either we have two domains: the Urdomain for the variables a; b and the integers for the variable n. But then the operation of subtraction is only defined for the Urdomain, yet not for 1, 0 and the negative numbers. (ii) Or we have one domain for all three variables, the positive and negative integers including 0, but then we have to expand the definition of subtraction to the entire domain of integers such that also the subtraction of negative numbers is included. The only reasonable alternative seems to be to define the whole domain of integers and the operation of subtraction in one fell swoop. But before we take such a bold step let us better think twice and reflect, what the reason for the difficulty might be. The deeper reason for the gap has not only to do, as we have seen, with the fact that the Urdomain is too small, but also with the arithmetical operations themselves, because these lead inevitably outside the Urdomain. It is therefore interesting to see that from Husserl’s point of view the gap is primarily caused by a lack of understanding of the arithmetical operations as “formal” calculus devices – what they mean with respect to the Urdomain – and not so much by the limited set of numbers of the Urdomain. In order to appreciate Husserl’s answer one has to consider the question, which operations can be performed completely “inside” the Urdomain. With respect to this question Husserl’s answer is firm and definite. There are only two operations, which can be executed inside the Urdomain of


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immediately given numbers: (i) the concatenation [Verbindung] of two numbers to another number; (ii) the separation [Teilung] of one number into two, not necessarily equal numbers.18 It is important to note that these two operations are basically different from the formal arithmetical operations of addition and division. The latter cannot even be defined in terms of the former, because it is in one respect more general, in another more special. Take for example the division n=m. It is on the one hand a special case of a separation – namely the separation of n into m equal parts. But on the other hand in arithmetic this operation is defined much more generally than it could ever be executed in the Urdomain, where only integral multiples of m can be divided by m. Therefore, the conclusion seems inevitable that the gap can’t be bridged. And this is precisely what Husserl believed for at least nine/ten years after he had finished PA.19 But then, about 1900, things changed. There seemed to be a way out of the difficulties. And this way has much to do with Hilbert’s axiomatic method in distinction to Husserl’s genetic approach to arithmetic. The main idea of the solution has, utterly simplified, two parts: (i) instead of generating the numbers step by step, we introduce the whole of arithmetic in one fell swoop, that means in particular we introduce the basic arithmetical operations like addition and subtraction, multiplication and division, etc. together with all necessary individual elements, such that they match each other; (ii) we prove that the resulting theory is deductively consistent, that means, we demonstrate that it is impossible to derive a contradiction from the theory in question. Now, you know, this solution has many malaises, but its principal defect in the present context is that it’s absolutely unintelligible why it is a solution to the original problem at all. Remember, the original problem was, how we can close the gap between the two levels of arithmetic, the formal calculus and its phenomenological basis. But now, the basis is not even mentioned in the supposed solution. How can a proof of the consistency of the “formal” calculus close the gap between the two levels? That seems impossible. Here, I must confess I did not understand for a long time what was going on. Something crucial was missing in the pretended solution. And indeed: The solution, that I have just stated above, is not Husserl’s solution, but Hilbert’s program of a proof-theoretical foundation of arithmetic, as outlined for the first time in 1900 at the Mathematical Congress in Paris. Husserl’s own, original proposal looks rather differently and, as we will see in a moment, much more appropriate to close the gap. Now, my strategy for the rest of the paper will be the following. First, I will explain as precisely as possible Husserl’s original solution as it is outlined in the draft of the Doppelvortrag [and some closely related frag-

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ments]. This is not an easy task, because Husserl was not very clear in stating his solution, to say the least, and sometimes even literally inconsistent. In order to grasp his solution one has therefore to interpret him according to the “principle of charity”; at least that is roughly, what I will do. Having done this, I will compare Husserl’s solution with Hilbert’s axiomatic approach and ask how far it is an answer to Husserl’s initial problem.

3. HUSSERL’S “WAY OUT” To begin with I have to assure you that Husserl in 1901 is still on the same epistemological trail, as he was all the time since he wrote PA. This is easily done. Let me just quote the passage in which he states the main problem of the Doppelvortrag. After having made some cryptical remarks concerning the relation of mathematics and philosophy in the foundations of arithmetic he comes to the main problem, the problem of the Imaginary, as he calls it: A domain of objects shall be given, in which, by the particular nature of the objects, forms of connections and relations are determined, that can be expressed in a certain system of axioms A. On account of this system, hence, on the basis of the particular nature of the objects certain forms of connections have no real meaning (reference), that is to say they are counterintuitive forms of connections. With which justification is it legitimate to use the counterintuitive in calculations, with which justification can the counterintuitive be used in deductive thinking, as if it were concordant. How can it be explained, that one can operate with the counterintuitive according to rules, and that the resulting sentences are true, if the counterintuitive has dropped out.20

I suppose that even a native speaker would have difficulties to understand precisely what Husserl is saying here. Therefore let me briefly explain what he alludes to: It’s the old problem in new and more general terms: How can we close the gap between a first given domain of objects and its description by an axiom system A and a second, formal deductive system B, that entails “counterintuitive” (widersinnige) operations which have no real meaning in the given domain of objects, but lead nevertheless to true results, once the counterintuitive terms have been removed from the descriptive sentences. It is important to note that Husserl does not put the use of a formal deductive system and its counterintuitive operations in question, but instead asks for a justification of their use. In this respect Husserl’s attitude to the practice of mathematics is almost identical to that of Hilbert, who also wants to save Cantor’s paradise of set theory in spite of all justified doubts about its consistency. The real difference between Husserl and Hilbert is another one. While Husserl stresses the epistemological significance of the gap


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between the different levels of arithmetic, Hilbert tries to downplay it. But this is, to be sure, only a difference in style, not in principle, because, as we will see, Hilbert acknowledges eventually the epistemological difficulty regarding a well founded justification of arithmetic. But then, in the twentieth, he is, of course, convinced that he can overcome the difficulty by finite means. But let’s return to Husserl and see what his new “solution” looks like. After a critical assessment of five proposals how to deal with the imaginary, which I have not the time to go into, although they are highly interesting, in particular Husserl’s rejection of Dedekind’s creative definitions and Cantor’s set-theoretical analysis of the continuum, Husserl turns to his own proposal. In first approximation, this proposal is quite the same as Hilbert’s program, that I had already indicated above, however, with one important difference. Because this difference is absolutely crucial, let me first explain it in modern terms, before I give you Husserl’s own formulation and then try to convince you that my interpretation is right. The principal idea is the following: If we have two systems of axioms, A and B , such that B is an extension of A, and furthermore, if B is consistent and A is complete, then each and every consequence of B , that can be expressed exclusively in terms of A, is “true”. Now, let us ask: Why is that a solution of the old problem? Why does it close the gap between the two levels of arithmetic? A first answer, I think, is simple and straightforward. Suppose A is an axiom system of the Urdomain of numbers including the two intuitive operations of concatenation and separation. Suppose furthermore, first that A is consistent, and second that A is complete. The possibility of the latter may be doubted, but because it is decisive, let’s assume it at least for the sake of argument. Then each and every sentence, expressed exclusively in the language of A (including some of the logical relations), is either itself a consequence of A, or it contradicts A, i.e. its negation is among the consequences of A. That’s just the standard meaning of “deductive completeness”. Consequently, if the axioms of A are true with respect to the Urdomain, then the consequences of A are also true with respect to this domain. That is the truth-conserving-principle of logical deduction. This in turn implies that every sentence, expressed in the language of A, is either true or false according to its membership in one of the two complementary classes of sentences, the consequences of A or the nonconsequences of A. Now suppose, that B is an axiomatic extension of A, that is deductively consistent [but not necessarily complete]. This implies not only that no sentence of the form S and non-S is deducible, but also that every consequence of B is compatible with [in the sense of

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not contradicting] any consequence of A. This in turn means that every consequence of B , that can be expressed exclusively in terms of A, is itself a consequence of A and for this reason true with respect to the Urdomain. Let us pause for a moment and ask again, in which sense is this proposal a solution of the old problem, (if it’s a solution at all)? In which precise sense does it close the gap between the different levels of arithmetic, the formal and the intuitive one? The second answer is: only in a rather restricted and conditional sense. It certainly does not close the gap in the sense of a logical reduction of the counterintuitive operations to the intuitive ones; it doesn’t define “subtraction”; “division”, “roots” etc. In terms of concatenation and separation of positive integers. No logical trick can accomplish the impossible! But it does close the gap in the sense of a justification! That means: it legitimates the use of the counterintuitive operations in the following way. First, the set of consequences of A and, hence, of true sentences with respect to the Urdomain [expressed in the language of A] is not changed by the use of counterintuitive operations – it becomes neither diminished nor increased. [The latter aspect is particularly important, because we tend to think otherwise. But, if A is complete, there is not a single consequence of B , that is expressed exclusively in terms of A, that cannot be deduced from A, regardless how difficult the deduction may be.] Second, and intuitively more important, all counterintuitive operations lead, once they are brought back into the Urdomain, to concordant results within the Urdomain. Let me explain the notion of concordance more closely, because it is crucial for the legitimate use of counterintuitive operations. Take again the operation of subtraction in the domain of natural numbers: if a > b, then a b is defined as: 9n(b + n = a). But what, if a < b? We have said above, then a b is undefined. But this is not the end of the story. We can consistently proceed in the following way: if a < b, then we take a b auxilary as the absolute value of b a, in symbols: jb aj. If we then come across the same operation in another context, let us say d + (a b), we take this as meaning d minus the absolute values of a b, in a formula d jb aj. Now, we have again two possibilities: Either d > jb aj, then d + (a b) [which is equal to d jb aj] is defined as before: 9n(jb aj + n = d). Or d < jb aj [as before], then we repeat the same procedure and stipulate a new auxilary result jd jb ajj. Now, it is easy to prove that this procedure, this excursion into the imaginary by means of auxiliary symbols, leads to the same result as the determination of d + (a b), first by concatenating a and d, and then subtracting b from the sum (d + a). This last operation, to be sure, is defined in the same


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instances as before, because (d + a) > b just in case d > jb aj; in the other instances we have to make another excursion into the imaginary. I have elaborated this mini example of a counterintuitive operation so painstakingly not only in order to demonstrate that such operations can be used consistently, of this we are convinced anyway, but also that their use is justified through their exceptionless concordance with other operations in the Urdomain. Furthermore the example shows that the counterintuitive operations can be partially reduced to the intuitive ones, that is to concatenation and separation of immediately given numbers, if we grant ourselves certain auxiliary symbolic expressions like negative or absolute numbers as jb aj. But, of course, a partial reduction is not a complete reduction in the logical sense of the term; an important part of the total meaning of the operation in question is missing in the reductum: the subtraction of negative numbers. Against this justificational approach, as I will call Husserl’s way out in distinction to a reductionistic program on the one hand and a pure axiomatic approach on the other, two major objections can be raised. First, it may be doubted, whether my charitable interpretation of Husserl is in agreement with his Doppervortrag. In particular one may ask, whether I am not too charitable in making Husserl’s conceptions and considerations logically more precise than they actually are. Indeed, there is another interpretation of the Doppelvortrag, to which I will come to at the end of the paper, that is much less charitable and accuses Husserl of having confounded in his metatheoretical notion of “definitness” two logically distinct concepts: the notion of deductive versus that of descriptive completeness. Second, in spite of Gödel’s incompleteness results for Peano Arithmetic, one may put Husserl’s assumption in question that the axiomatic description of the Urdomain is or can be made complete. First of all, no explicit system of axioms was presented. Hence, one cannot judge whether the axiomatic characterization of the Urdomain is complete. But without this assumption, the logical argument for the use of counterintuitive operations, leading into the imaginary, collapses. Furthermore, one can object that a complete axiomatic description of the Urdomain of numbers is not possible at all because genuine arithmetic is fundamentally incomplete. Therefore, Husserl’s whole idea of a justification is in vain. Let me take the second objection first. The first part of the objection is granted, but not the second: As far as I can see, Husserl presents nowhere an axiomatic description of the Urdomain, let alone a specification of the logic involved.21 Hence, we do not know whether such an axiomatic description is complete in the required sense that every sentence of the descriptive language is either true or false

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on behalf of the axiom system, that is to say that it’s either a consequence of the axioms or it contradicts them. It is therefore a legitimate and serious question, whether such a complete axiomatic description of the Urdomain of numbers exists.22 This question was in fact raised by Hilbert after Husserl’s talk in Göttingen.23 Hilbert’s question in Husserl’s own memo sounds such: “Had I [Husserl] been justified to say that every sentence entailing only the positive integers is either true or false on behalf of the axioms for the positive integers.” [Hatte ich recht zu sagen, daß jeder nur die ganzen positiven Zahlen enthaltende Satz aufgrund der Axiome für ganze positive Zahlen wahr oder falsch sei.]24 If we take the clause “that every sentence entailing only the positive integers is either true or false on behalf of the axioms for the positive integers” literally Hilbert’s question goes straight to the heart of the matter: namely whether an axiomatic, and that means a finite description of the domain of positive integers is possible, that is deductively complete. Of course, Husserl was convinced that it should be possible to give such an axiomatization. The reason for his conviction was, as far as I can see, embodied in the following argument: “every numerical equation [between positive integers] is true, if it can be transferred in an identity, otherwise false”.25 Consequently, if we restrict the axiomatization to the relations of numerical equality and inequality and the two operations of concatenation and separation of positive integers, a deductively complete axiomatization of the positive integers should be possible. Indeed this is roughly the way in which Hilbert later proceeded in order to prove the consistency of elementary arithmetic. The first axiom system proved to be consistent had only two descriptive relations: equality and inequality and one operation, namely addition, but no logical negation.26 This result shows that the second part of the objection can’t be completely correct: There could be deductively complete axiomatizations of arithmetic, Gödel’s result notwithstanding. Consequently, Husserl’s original idea to use the deductive completeness of an axiomatic description of the Urdomain in order to bridge the gap between the intuitively accessible numbers [positive integers] and formal arithmetic [with its imaginary numbers] was not as far fetched, as it appeared at first glance. Nonetheless there is something intriguing in Husserl’s original idea, that becomes disclosed by Gödel’s result: The direction, in which the completeness of the axiomatic description of the Urdomain is to be achieved, is exactly the opposite, in which the common sense of the word “complete” would induce one to go. Instead of enriching the description by more and more properties and relations, until it is complete in the common sense, the deductive completeness has to be achieved by thinning out the


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description until it entails only one pair of relations, that of numerical equality and inequality, and only one pair of operations, that of concatenation and separation [respectively addition and partition in the arithmetical sense]. This shows that the notion of deductive completeness is relative to the language, that is used in a description of a domain of numbers: the less concepts and operations are used, the easier is the deductive but, of course, not a descriptive completeness [in the common sense] achieved. Both notions go in opposite directions. Now, Gödel’s incompleteness result for Peano Arithmetic made it clear beyond doubt that there is an upper bound for the descriptive, complexity of number theory, beyond which no deductive completeness can be achieved. (Intuitively, the limit is reached if the theory permits diagonalization.) For this reason Gödel’s incompleteness result erects an upper limit for Hilbert’s program to prove the consistency of analysis by starting from rather “weak” and moving to “stronger and stronger” axiomatic theories of arithmetic. At this junction it is, however, important to note that Husserl’s idea to bridge the gap between weak and strong theories of arithmetic is not ruled out by Gödel’s incompleteness result. On the contrary, it is perfectly reasonable to search the “richest” axiomatic description, that is still deductively complete, and then to ask: How far can we go from this intuitively perspicuous theory towards analysis by semantically conservative extensions. This is roughly what is done in so called “weak theories” of arithmetic. The first objection [that my interpretation of Husserl is too charitable] is far more difficult to answer. A first answer, that shows that my “justificational” interpretation has at least some “textual basis”, is provided by the following quotation from the DV. After having considered at some length the intricate question how we come to know the truth of a sentence from the consistency of the corresponding axiom system, from which the sentence was deduced, Husserl presents the following meta-logical law: A passage through the imaginary is permitted: (1) if the imaginary can be defined formally in a consistent comprehensive system of deduction and if (2) the original domain of deduction has, formalized, the property that every sentence falling into this domain is either true on the basis of the axioms of this domain or false on their account, i.e. being in contradiction with the axioms.27

If you interpret the first condition as meaning an axiom system B, which is first consistent and second an extension of an original axiom system A, that in turn is complete on account of the axioms with respect to the Urdomain, you have precisely the solution discussed above. So much in defense of my charitable interpretation. But I have to admit that a different interpretation is possible, and that this interpretation is, at first glance, the more probable one. This inter-

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pretation puts greater emphasis on Husserl’s claim that his concept of “definiteness” is the same as Hilbert’s concept of “completeness”. But, I suspect, Husserl is in this case a victim of a confusion. Hilbert’s axiom of completeness, as it’s stated in the essay On the Concept Number as well as in the Foundations of Geometry, expresses something different than “deductive completeness”. It says, roughly, that a certain domain of objects, for example the real numbers or the points, straight lines and planes of Euclidean geometry, cannot be expanded by new objects – “intruders” so to speak – under maintaining all the axioms established so far as valid in the domain of given objects. In modern terms, Hilbert’s axiom of completeness asserts28 the categoricity of an axiom system – not its deductive completeness! However, it is important to remark that Husserl was by no means the only victim of this confusion. Indeed ironically enough, Hilbert himself was not quite clear about the difference between the two metalogical notions. Not only did he characterize the concept of completeness as the property of an axiom-system to be sufficiently strong to prove all [geometrical] sentences,29 but he also tried to wed both notions together, when as late as 1928 he maintained30 that the usual categoricity-criterion for the completeness of number theory – the fact that any two models of number-theory are isomorphic to each other – is too weak from the finite point of view and has to be replaced by the proof that every formula , consistent with the axioms of number-theory, is itself “provable”. The last notion is, however, nothing but the old deductive completeness in disguised form. This shows how difficult it was, even for a mathematical genius like Hilbert, to grasp these meta-logical notions clearly and distinctly.

P

3.1. Postscript After I had finished this essay, another contributor to this colloquium, A. Kanamori, brought to my attention an essay by Claire Ortiz Hill, that has almost the same title. Although the topic of the two essays is obviously the same – Hilbert and Husserl on Completeness – the point of view from which they are written is rather different. Anybody, interested in Husserl and Hilbert and the emergence of the meta-theoretical concept of “completeness”, should read both, and make his own judgement.

NOTES 1

This danger is quite analogous to what Dummett and most philosophers of language have done to Frege himself, namely to read and interpret him through the glasses of L. Wittgen-


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stein, in particular through those of the Tractatus. However, Frege was no philosopher of language, at least not in the first place, but a classical mathematician. 2 There are, of course, some minor exceptions, such as J. P. Miller’s book “Numbers in Presence and Absence”, pp. 125–29, M. Nijhoff (1982), J. Scanlon’s essay “Tertium Non Datur: Husserl’s Conception of a Definite Multiplicity” (1991), as well as Lothar Eley’s “Einleitung” to Husserl’s PA, Husserliana-Vol. XII. All miss, however, in my view, the main point, the proper significance, of this work. 3 Husserl became by Ministerial-Bestallung of September 14. 1901 “ausserordentlicher Professor” with a “Lehrauftrag für das Gesammtgebiet der Philosophie”. 4 See the “Einleitung” to Husserliana XII by Eley, in which this point is likewise stressed and most of the relevant remarks are quoted. 5 The exact dates for Husserl’s lectures were unknown so far. The closest was Husserl’s own hint “November 1901”. 6 See the first of the three “Beilagen” to the DV, Husserliana XII, p. 445. After having summarized Hilbert’s lecture in three short statements Husserl continues: “Hilberts Einwand. –” As the subsequent exposition of “Hilberts Einwand” (in Husserl’s own presentation) makes clear, this objection can only mean his own, Husserl’s position. 7 ¨ For Husserl’s acquaintance with Hilbert’s essay “Uber den Zahlbegriff” we have only indirect, yet sufficient evidence. Husserl says in a footnote in Ideen, p. 168, referring to his DV: “Die nahe Beziehung des Begriffes der Definitheit zu dem von D. Hilbert für die Grundlegung der Arithmetik eingeführten “Vollständigkeitsaxiom” wird jedem Mathematiker ohne weiteres einleuchten”. This can only mean that Husserl did know Hilbert’s essay, because in 1901 there is no other place he could refer to. 8 These notes are now published as “Beilage I” to appendix VI in Husserliana XII (PA). 9 For a critical assessment of Frege’s and Hilbert’s debate about the foundations of geometry see my preprint “Logic and Geometry”, TUM, Mathematisches Institut, (1997). 10 This presentation of Husserl’s PA is in itself an interesting and remarkable story, because it shows that Weyl had already a deep sympathy for Husserl’s phenomenological approach, much earlier than it is usually supposed. But I have no time to go into this. 11 “Der Begriff der Menge und Anzahl hat während der Entwicklung des menschlichen Geistes verschiedene Stufen durchlaufen. Auf der ersten Stufe handelt es sich um die eigentliche Inbegriffsvorstellung, welche zustande kommt, wenn Vorstellungen mehrerer für sich bemerkter Einzelobjekte durch ein einheitliches Interesse aus unserem Bewußtseinsinhalt herausgehoben und zusammengehalten werden. Auf dieser Stufe bezeichnen die niedrigsten Zahlwörter, sagen wir 2, 3 und 4, unmittelbar merkliche Unterschiede des bei ¨ der Inbegriffsvorstellung in Funktion tretenden psychischen Aktes:” Uber die Definition mathematischer Grundbegriffe, Ges. Abh. Bd. I, 302. 12 “Auf der zweiten Stufe treten für die eigentlichen Vorstellungen symbolische ein. Als das bedeutendste Erzeugnis dieser zweiten Periode hat das bekannte, jedem Kind geläufige symbolische Zählverfahren zu gelten, welches gestattet, auch inhaltreichere Mengen nach ihrer Anzahl zu unterscheiden. Bei seiner Ausbildung spielt wohl auch ein gewisses Möglichkeitsgefühl eine wichtige Rolle, indem wir uns, um der Außenwelt gerecht zu werden, nicht an die zufälligen Beschränkungen und Mängel unserer Sinnesorgane und geistigen Fähigkeiten gebunden fühlen.” – (ibid.) 13 Daraus, daß eine eigentliche Vorstellung unendlicher Mengen in dem Sinne, daß die einzelnen Elemente derselben als für sich bemerkte Inhalte in unserem Bewußtsein gleichzeitig gegenwärtig sind, nicht vollziehbar ist, kann ebensowenig ein Einwand gegen ihre logische Zulässigkeit erhoben werden, wie dies bei endlichen aus einer größeren Anzahl von Elementen bestehenden Mengen geschehen kann, die doch gleichfalls nicht eigentlich

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vorstellbar sind; und nur in diesem Sinne der Unmöglichkeit eines eigentlichen Vorstellens unendlicher Mannigfaltigkeiten ist es wahr, wenn man sagt: ein Aktual-Unendliches gibt es nicht. 14 Da wir durch andere unabweisbare Gründe – die Analysis zwingt uns dazu – genötigt sind unendliche Mengen einzuführen, handelt es sich schließlich auf der dritten Stufe darum, die Theorie der endlichen und unendlichen Mengen und Zahlen in einer wissenschaftlichsytematischen Weise durch Aufstellung von Axiomen, Definitionen und daraus gezogenen Folgerungen aufzubauen. Bei diesem Aufbau werden wir nach dem vorhin Ausgeführten kein Bedenken tragen, das eigentlich intendierte Begriffssystem durch andere ihm vollständig isomorphe zu ersetzen.” 15 For lack of a better expression I call these numbers the first numbers, although no ordering is intended by this expression. Furthermore, for the sake of brevity, I don’t distinguish between the abstract numbers II, III, IIII, : : : and their numerals 2, 3, 4, : : : , if not necessary. In other words, I take the numerals 2, 3, 4, : : : as “rigid designators”. 16 The Urdomain of immediately accessible numbers entails only the numbers 2 : : : 12 because, according to psychological experiments of those days, we can at best present 12 elements simultaneously in our consciousness. This is one of several psychological moments of Husserl’s PA, that later became so vehemently criticized by Frege and – in spite of this – eventually abandoned by Husserl himself. 17 In the subsequent discussion I am following Weyl’s proposal to define subtraction in two ¨ steps. See Weyl “Uber die neue Grundlagenkrise der Mathematik”, GA, II, pp. 160. 18 See PA, p. 190. It should, however, be added that the above statement is not quite correct, because the separation of number 2 leads to the number 1, which is no member of the Urdomain. Similarly, sometimes the concatenation of two numbers of the Urdomain, let’s say 7 and 8, leads to numbers outside the original Urdomain. Consequently, the above statement is only correct if we take the entire set of natural numbers as the Urdomain. In the rest of the paper it is supposed that this “expansion” of the original Urdomain has already been made. 19 For the sake of clarity I should add that Husserl’s final goal was, of course, the closure of the gap. And in fact, after the publication of PA he made several attempts to achieve that. But he was too much a mathematician not to see that all these attempts were not successful. Particularly revealing in this respect is his lecture in 1895 about “neuere Fortschritte in der deduktiven Logik”: There he confesses quite clearly – after having discussed three different methods to introduce and justify the calculus of arithmetic – that the operations of subtraction and division have no validity whatsoever in the Urdomain of positive integers [Anzahlen]. Let me quote from Husserliana XII, p. 68. “In der Lehre von den Anzahlen hat der Begriff a b keine Geltung, wenn b > a ist. 3 7 ist eine unmoegliche Zahl, ich kann nicht 7 Einheiten von 3 wegnehmen, ich erhalte durch die geforderte Operation keine Zahl. Anders verhaelt es sich hier (im Gebiet des arithmetischen Kalkuels der sog. Streckenrechnung): Ich erhalte eine Zahl, nämlich eine negative Zahl. Das Gebiet der jetzigen Streckenzahlen ist also ein weiteres: Die eine Hälfte mitsamt den dafür geltenden Sätzen deckt sich mit dem Anzahlengebiet. Die andere Hälfte ist u¨ berschüssig, es fehlt das Korrespondierende im Anzahlgebiet. – Auch noch in anderer Beziehung ist das Gebiet der Streckenzahlen, und zwar sowohl der einseitigen als der zweiseitigen unendlichen Geraden, umfassender. Im Gebiet der Anzahl hat der Begriff 3/7 keine Geltung. Ich kann nicht die Zahl 3 in 7 Teile teilen, deren jeder wieder eine Zahl ist, wie doch im Begriff der Division verlangt ist. Aber sehr wohl kann ich eine Strecke der Länge 3 in 7 gleiche Teile teilen, und ich kann solch einen Teil wieder zahlenmässig auffassen, indem ich die sogenannten Bruchzahlen einführe, 3: 7 = 3 1/7, 1/7 eine Strecke

f

g


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bedeutet, die den siebten teil der Strecke 1 darstellt, oder die Strecke, die versiebenfacht 1 gibt. Eine Anzahl 1/7 ist etwas Absurdes. So ist das Gebiet nun unendlich viel reicher. Es existiert die Brucheinheit 1/2, 1/3, 1/4, 1/5, : : : und alle daraus abgeleiteten Zahlen.” 20 Es sei ein Gebiet von Objekten gegeben, in welchem durch die besondere Natur der Objekte Verknüpfungs- und Beziehungsformen bestimmt sind, die sich in einem gewissen Axiomensystem A aussprechen. Aufgrund dieses Systems, also aufgrund der besonderen Natur der Objekte haben gewisse Verknüpfungsformen keine reale Bedeutung, d. h. es sind widersinnige Verknüpfungsformen. Mit welchem Recht darf das Widersinnige rechnerisch verarbeitet, mit welchem Recht kann also das Widersinnige im deduktiven Denken verwandt werden, als ob es Einstimmiges wäre. Wie ist es zu erklären, daß sich mit dem Widersinnigen nach Regeln operieren läßt, und daß, wenn das Widersinnige aus den Sätzen herausfällt, die gewonnen Sätze richtig sind. 21 The habit of using logic silently without explicitly stating its laws and rules is typical for most mathematicians of the time. In this respect Husserl is in good company. We find the same habit in Pasch and Hilbert’s Foundations of Geometry. Nonetheless, for the question of completeness this habit is disastrous. 22 As the Urdomain of numbers I take here – in agreement with Husserl – the positive integers. This means that the original Urdomain of immediately accessible numbers, the positive integers 2, 3, : : : up to 12, has already been expanded twice in order to be able to perform the operations of separation and concatenation unlimitedly. 23 See the Introduction, page 4, and in particular footnote 1 on page 4. 24 PA, Husserliana XII, VI. Abhandlung, “Das Imaginäre in der Mathematik”, p. 445. 25 “Jede numerische Gleichung ist wahr, wenn sie sich in eine Identität u¨ berführen läßt, sonst falsch”, PA, p. 443. 26 See Hilbert [1922] “Neubegründung der Mathematik”, Gesammelte Abhandlungen, Band III, p.173; Hilbert’s six axioms for elementary arithmetic are the following: 1. a = a 2. 1 + (a + 1) = (1 + a) + 1 a+1 = b+1 3. a = b 4. a + 1 = b + 1 a=b 5. a = c (b = c a = b) 6. a + 1 = 1. 27 PA, Husserliana XII, p. 441 Ein Durchgang durch das Imaginäre ist gestattet: (1) wenn das Imaginäre sich in einem konsistenten umfassenden Deduktionssystem formal definieren läßt undwenn (2) das ursprüngliche Deduktionsgebiet formalisiert die Eigenschaft hat, daß jeder in dieses Gebiet fallendeSatz entweder aufgrund der Axiome dieses Gebietes wahr oder aufgrund derselben falsch, d. i. mit den Axiomen widersprechend ist. 28 That Hilbert asserts the categoricity of Euclidean geometry instead of proving it, is a strange feature of his idea, that was already criticized by Husserl. 29 ¨ See Hilbert’s early essay “Uber den Zahlbegriff” [1900], where he says: “Es entsteht dann die notwendige Aufgabe, die Widerspruchslosigkeit und Vollständigkeit dieser Axiome zu zeigen, d. h. es muß bewiesen werden, das die Anwendung der aufgestellten Axiome nie zu Widersprüchen führen kann, und ferner, daß das System der Axiome zum Nachweis aller geometrischen Sätze ausreicht” (my emphases). 30 See problem IV and V in his “Problems in the Foundations of Mathematics”.

6

! !

! !

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REFERENCES

¨ Hilbert, D.: 1900, ‘Uber den Zahlbegriff’, Jahresbericht der DMV, Bd. 8. Hilbert, D.: 1928, ‘Probleme der Grundlegung der Mathematik’, Congresso Internationale dei Matematici, Bologna, 3–10 Settembre 1928, Zanichelli (ed.). Hill, C. O.: 1995, ‘Husserl and Hilbert on Completeness’, From Dedekind to Gödel, Synthese Library Vol. 251, Jaakko Hintikka (ed.). Husserl, E.: 1891, Philosophie der Arithmetik, quoted from the Husserliana-Edition Vol. XII. Majer, U.: 1997, Logik und Geometrie, Preprint des Mathematischen Institutes der Technischen, Universität München, TUM. Universität Göttingen Institut für Wissenschafts geschichte Humboldtallee 11 37073 Göttingen Germany [email protected]