This is a preprint version of the paper that appeared in definitive form in From Dedekind to Gödel, Jaakko Hintikka (ed), Dordrecht: Kluwer, 1995. It is also published in Husserl or Frege? by Claire Ortiz Hill and Guillermo Rosado Haddock. The published versions should be consulted for all citations.

CLAIRE ORTIZ HILL

HUSSERL AND HILBERT ON COMPLETENESS*

In a 1900 paper entitled "On the Number Concept", the formalist mathematician David Hilbert proposed a set of axioms from which he hoped arithmetic might be derived. The last of these axioms was an "Axiom of Completeness" stipulating that: "It is not possible to adjoin to the system of numbers any collection of things so that in the combined collection the preceding axioms are satisfied; that is, briefly put, the numbers form a system of objects which cannot be enlarged with the preceding axioms continuing to hold.'' l

In his major works2 the philosopher Edmund Husserl wrote that he had appealed to a concept of completeness closely related to the axiom of completeness Hilbert had introduced for the foundations of arithmetic. In these works Husserl is specifically referring to his theory of complete manifolds (Mannigfaltigkeiten) which, as he wrote in Ideas §72, have the "distinctive feature that a finite number of concepts and propositions --to be drawn as occasion requires from the essential nature of the domain under consideration-- determines completely and unambiguously on the lines of pure logical necessity the totality of all possible formations in the domain, so that in principle, therefore, nothing further remains open within it." In such complete manifolds, Husserl maintained, "the concepts true and formal implication of the axioms are equivalent."

It is clear from Husserl's writings that he considers the fact of this kinship to be quite significant. And in § 31 of his 1929 Formal and Transcendental Logic, Husserl even went so far as to say that the close study of his analyses would reveal that the underlying, though inexplicit, reasons which had led Hilbert to attempt to complete a system of axioms by adding a separate axiom of completeness were much the same as those which had played a determinant role in Husserl's own independent formulation of his concept of completeness.

The few people who have commented on Husserl's remarks have principally tried to determine whether or not Husserl's concept was in fact the same as Hilbert's, and a few have considered the relevance of Husserl's remarks to issues in mathematical logic.3 Considering, however, what Gödel's theorems have brought to the subject, and given the radically different contexts into which Husserl and Hilbert integrated their ideas on completeness, I consider those particular questions to be rather academic. Since Husserl was ultimately inquiring into the foundations of all knowledge his notion played a role in investigations which were vastly broader in scope and essentially different in nature than Hilbert's inquiries into the foundations of mathematics were.

What intrigues me rather is Husserl's belief that, though inexplicit, Hilbert's deep underlying reasons for formulating his axiom of completeness were basically the same as those which had led Husserl to formulate his own concept of completeness. Unfortunately, however, Husserl was not himself very explicit about the steps in his own reasoning which had led to the formulation of his views on completeness, and he seemed to think the connection between his idea and Hilbert's was self-evident.4

Here I want to inquire further into the origins of Husserl's ideas on completeness, and then look at how Husserl thought he might provide more secure logical foundations for all knowledge by generalizing insights drawn from his investigations into the foundations of mathematics. Specifically, I argue that early in his philosophical career Husserl came to believe that having had recourse to the ideal elements he thought were necessary for the kind of foundations for knowledge he sought was justifiable when these elements admitted of formal definition within a complete deductive system. I draw Frege's ideas into the discussion because Frege tangled with the very same problems in the foundations of arithmetic that led Husserl to ally himself with formalism.

1. The Foundations of Mathematics and Imaginary Numbers

Husserl's interest in the foundations of mathematics has a noble ancestry, the full account of which has yet to be written and cannot be told here. Mathematically speaking, Husserl had the good fortune to be in the right place at the right time, and so was personally on hand to witness important developments in mathematics which Russell, Whitehead, Frege and Wittgenstein mainly knew by description.

After studying under Karl Weierstrass and serving as his assistant,5 Husserl studied for a time under Franz Brentano, the philosopher working to reform Aristotelian logic whose ideas on intentionality would later be so instrumental in freeing Bertrand Russell from the bonds of subjective idealism.6 In 1886 Husserl moved to the University of Halle where for the next fourteen years he maintained close professional and personal ties with Georg Cantor.7 Weierstrass, Brentano and Cantor all three worked on the ideas of Bernard Bolzano,8 one of the pioneers in the area of completeness.9

Husserl's interest in completeness and formalist foundations for mathematics is traceable back to those early years in Halle, and as he told his readers on several occasions, originally derived from problems regarding imaginary numbers which first came up while he was trying to complete his 1891 Philosophy of Arithmetic.10

The matter arose as he searched for answers to questions regarding the consistency of arithmetic, and it especially involved his attempts to account for the achievements of certain purely symbolic procedures of mathematics despite their appeal to apparently nonsensical combinations of symbols. Husserl sought for a way to explain, or to explain away the many expressions which appear in philosophical and mathematical reasoning but which do not, and cannot designate objects.

Husserl's contemporary Alfred North Whitehead provides some insight into how mathematicians of the time viewed the problem which Husserl cited as the source of his views on completeness. Problems concerning imaginary numbers, Whitehead informed readers in 19ll,11 arose from the consideration of quadratic equations like x2 + 1 = 3, x2 + 3 = 1 and x2 + a = b. The first equation, he explains, becomes x2 = 2, and this has two alternative solutions, either x = + v2, or x equals - v2. And this is where the problem begins. For the equation x2 + 3 = 1 yields x2 = -2, but there is no positive number which when multiplied by itself will yield a negative square. So, if the symbols mean ordinary negative or positive numbers there is no solution to x2 = - 2, which must then be nonsense. One cannot therefore say that the symbols may mean numbers and "a host of limitations and restrictions" begin to accumulate. A new interpretation of such symbols is therefore required so that negative squares might have meaning. It was ultimately perceived, Whitehead writes, how convenient:

it would be if an interpretation could be assigned to these nonsensical symbols. Formal reasoning with these symbols was gone through, merely assuming that they obeyed the ordinary algebraic laws of transformation; and it was seen that a whole world of interesting results could be attained if only these symbols might legitimately be used. Many mathematicians were not then very clear as to the logic of their procedure, and an idea gained ground that, in some mysterious way, symbols which mean nothing can by appropriate manipulation yield valid proofs of propositions.

Faced with this problem, Husserl said his main questions were: 1) Under what conditions can one freely operate within a formally defined deductive system with concepts which according to the definition of the system are imaginary and have no real meaning? 2) When can one be sure of the validity of one's reasoning, that the conclusions arrived at have been correctly derived from the axioms one has, when one has appealed to imaginary concepts? And 3) To what extent is it permissible to enlarge a well defined deductive system to make a new one that contains the old one as a part? He eventually concluded that if the system was complete, then calculating with imaginary concepts could never lead to contradictions.12

Such questions about the logical foundations of the real number system led Husserl to want probe more deeply in order to clarify its structure. They were also, he tells us, instrumental in undermining his faith in Brentano's psychologism, which Husserl came to realize could not provide the real number system with the sound foundations he sought. For him these questions were also linked with deep frustrations he felt with regard to Kant's concept of analyticity which Husserl thought was too weakly formulated and in dire need of reform.13 By early 1891 Husserl had in fact become convinced of the necessity of providing knowledge with a strong, formal, scaffolding in the Leibnizian sense of a mathesis universalis which would act as a guarantor of objectivity and be a safe bulwark against the incursions of psychologism or subjective idealism.14

It was in 1891, the year in which Husserl published the Philosophy of Arithmetic, that we first find clues that he was looking to formalist theories of arithmetic as a way of avoiding onerous problems with imaginary numbers and of establishing consistency in arithmetic. During that year he was working on the never to be published second volume of the Philosophy of Arithmetic.l5 In it he planned to deal with the fractions, negative, and irrational numbers that he included under the heading of the imaginary.16 Reading his 1890 drafts one is witness to his mounting frustration, and even anger, concerning the lack of any clear, logical understanding of the way in which mathematicians use such numbers, and then to the new confidence he finally won in 1891 in declaring arithmetic to be an analytic a priori discipline.17

That year Husserl wrote a letter to Carl Stumpf explaining his disillusionment with Brentano's methods and his new faith in the arithmetica universalis, which Husserl now thought of as a part of formal logic defined as a technique of signs making up a special, important chapter in logic as theory of knowledge. His new views, he wrote, would require important reforms in logic and he knew of no logic that would even give an account of the very possibility of a genuine calculational technique.18

So Husserl's earliest ideas on completeness were tied in not only with his inquiries into the logical foundations of the real number system, but also tied in with a more specifically philosophical quest to clarify the sense of the analytic a priori and develop a pure analytic logic free of any taint of psychologism. They reach deep into his reasons for abjuring Brentano's psychologism.

2. In Conflict with Frege

Husserl's earliest ideas on completeness also reach deep into his reasons for shunning Frege's logic, and so a good look at the connections between Husserl's ideas and formalist theories of mathematics actually provides insight into the nature of Husserl's clash with Frege, and sets into perspective the charge frequently leveled at Husserl that he lapsed back into psychologism not long after having repudiating it.

Husserl thoroughly studied Frege's Foundations of Arithmetic in the first volume of Philosophy of Arithmetic, sharply attacking the theory of extensionality and identity Frege espoused there.19 Husserl never retracted those particular criticisms, but ten years later in the Logical Investigations he retracted the three pages of his criticism of Frege20 in which he had denied that one could provide sound foundations for arithmetic by deriving theorems from a series of formal definitions in a purely logical fashion.21

In an 1891 letter, Husserl wrote to Frege that he had only a rough idea of how Frege would justify the imaginary in arithmetic since the path Husserl himself had found after much searching had been rejected by Frege in his 1885 article "On Formal Theories of Arithmetic". In the passage of the article cited in Husserl's letter,22 Frege characterizes the unacceptable theory in question as one which but sets down rules by which one passes from the equations given to new ones in the way one moves chess pieces. Unless an equation contains only positive numbers, it no more has a meaning than the position of chess pieces expresses a truth. Now in virtue of these rules, Frege continues his criticism, an equation of positive whole numbers may actually appear. And if the rules are such that true equations can never lead to false conclusions, then only two results are possible: either the final equation is meaningless, or it has a content about which we can pass judgement. The latter will always be the case if it contains only positive whole numbers, and then it must be true, for it cannot be false. If the rules contain no contradictions among themselves, and do not contradict the laws of positive whole numbers, then no matter how often they are applied, no contradiction can ever enter in. Consequently, if the final equation has any meaning at all, it must be non-contradictory, and hence be true. This is a mistake, Frege concludes, for a proposition may very well be non-contradictory without being true.

The theory Frege has just opposed is apparently the one Husserl had come to favor. In his 1891 letter Husserl was not explicit about why he believed Frege's logic could not satisfactorily justify the imaginary in arithmetic. However, Husserl's reasons surely involved Frege's well-known thesis that in a logically perfect language, expressions that do not denote objects are unfit for scientific use. On his personal copy of Frege's article, Husserl particularly underlined the passage which reads: "The situation radically changes when one takes these figures to be signs of contents; in that case, the equation states that both signs have the same content. But if no content is present, the equation has no sense." In the margin Husserl wrote NB next to this passage.23

The use of signs or combinations of signs without reference was at the heart of Frege's dispute with formalists who, he believed, only manipulated signs without any regard for what those signs might stand for. In formal theories of arithmetic, Frege maintained, there is only talk of signs that neither have nor are meant to refer to objects. But, Frege insisted, "logic is not concerned with how thoughts, regardless of truth value, follow from thoughts" but "the laws of logic are first and foremost laws in the realm of reference . . . we have to throw aside proper names that do not designate or name an object ….24 Languages, Frege wrote in "On Sense and Reference":

have the fault of containing expressions which fail to designate an object (although their grammatical form seems to qualify them for that purpose) …. This arises from an imperfection of language from which even the symbolic language of mathematical analysis is not altogether free; even there combinations of symbols can occur that seem to mean something but (at least so far) do not mean anything, e.g. divergent infinite series…. A logically perfect language (Begriffsschrift) should satisfy the conditions, that every expression grammatically well constructed as a proper name out of signs already introduced shall in fact designate an object, and that no new sign shall be introduced as a proper name without being secured a meaning…. The history of mathematics supplies errors which have arisen in this way…. It is therefore by no means unimportant to eliminate the source of these mistakes.25

Frege explicitly allied himself with "extensionalist logicians". In my Grundlagen and the paper "Über formale Theorien der Arithmetik", he wrote,

I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs - e.g. Ö-1 has a reference or not, that, on the contrary, the whole cogency of the proof stands or falls with this. The reference is thus shown at every point to be thc essential thing for science . . . thc extensionalist logicians come closer to the truth in so far as they are presenting --in the extension-- a reference as the essential thing.26

Husserl, however, sided with Hilbert who wrote of how:

The method of ideal elements, that creation of genius, then allowed us to find an escape…. Just as i = Ö-1 was introduced so that the laws of algebra, those, for example, concerning the existence and number of the roots of an equation, could be preserved in their simplest form, just as ideal factors were introduced so that the simple laws of divisibility could be maintained even for algebraic integers (for example, we introduce an ideal common divisor for the numbers 2 and 1 + Ö -5, while an actual one does not exist) so we must here adjoin the ideal propositions to the finitary ones....27

Hilbert only stipulated one condition to which the method of ideal elements would be subject: "the proof of consistency; for extension by the addition of ideals is legitimate only if no contradiction is thereby brought in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain."28

Over the next several years Husserl worked hard to sort through the problems raised as he tried to finish his book on the philosophy of arithmetic. The result was his Logical Investigations, the first volume of which is the anti-psychologistic treatise, the Prolegomena to Pure Logic, in which Husserl formulated his views on completeness (§§ 69-70) in the way he would come to judge to be definitive (see Formal and Transcendental Logic, § 28). The Prolegomena was surely written in the late 1890s.29

In 1900 Husserl was appointed to Göttingen where he would stay for the next sixteen years. There he was warmly welcomed into Hilbert's circle.30 Husserl and Hilbert had much in common. Hilbert was just then lecturing on the calculus of variations (the subject of Husserl's doctoral thesis).31 Hilbert had also just posited his axiom of completeness for arithmetic. He was an admirer of Weierstrass and Cantor.32 Moreover, Hilbert was also just then corresponding with Frege on the very matter of truth and logical consistency at issue in the Frege passage I summarized above. As long as he had been thinking, writing and lecturing on these things, Hilbert wrote of himself to Frege in late 1899, he had been saying that if the arbitrarily given axioms did not contradict one another, then they were true. This conception, he wrote, was the key to understanding his own recent work on the axioms of arithmetic and geometry, and the talk of completeness to be found there. Husserl had access to the Frege-Hilbert correspondence and partial copies of it, along with notes Husserl made on it which were found in his Nachlass.33

3. Completeness and the Imaginary

Invited by Hilbert and Felix Klein, Husserl addressed the Göttingen Mathematical Society in 1901 on the subject of completeness and the imaginary.34 At the highest level, he told his audience, mathematics is the science of deductive systems in general. By appealing to a set of formal axioms which are consistent, independent, and purely logical in that they obey the principle of non-contradiction, it yields the set of propositions belonging to the theory defined. However, methodological questions arise when one tries to apply these formal techniques to the real number system and to specific domains of knowledge. These questions are a matter of serious concern to philosophers because their understanding of the general nature of the deductive sciences, and of theories in general, depends on their being able to resolve them. The development of the sciences, Husserl warned his listeners, had constantly shown that lack of clarity in the foundations ultimately wreaks its vengeance, that if certain levels of progress are reached, further progress is fettered by errors due to obscure methodological ideas. (Husserl 1901, 431-32)

Questions regarding imaginary numbers, he continued, had come up in mathematical contexts in which formalization yielded constructions which arithmetically speaking were nonsense but which, astonishingly, could nonetheless be used in calculations. It became apparent that when formal reasoning was carried out mechanically as if these symbols had meaning, if the ordinary rules were obeyed, the results did not contain any imaginary components, then these symbols might be legitimately used. And this could be empirically verified (p. 432).

Husserl did not believe that general logic could shed light on the mystery because of the importance logicians accord to working with clear, precise, unambiguous concepts so that contradictions do not sneak in unnoticed. Logicians would ban contradiction, he said. For them contradictions only serve to show that a concept does not have an object, and contradictory concepts but yield contradictory consequences to which no object will correspond. But with the imaginary in mathematics that is plainly not the issue (p. 433).

Imaginary numbers may be countenanced, Husserl concluded, when they admit of formal definition within an enlarged consistent deductive system, and when the original formalized domain of deduction has the property that any proposition within the domain is either true on the basis of the axioms of that domain, or false, meaning in contradiction with the axioms. In addition, Husserl maintained, one needs to be clear about what is meant by a proposition's being in the domain. This, he argued, can only be determined if one can tell beforehand whether propositions deduced from the larger domain are situated in this sense in the more restricted domain and this can only be known if one knows from the outset that the proposition falls under this axiom in this sense. This is possible in so far as the axioms determine the domain completely, in so far as no other axiom may be added (p. 441).

Once in Hilbert's company Husserl did not slavishly copy Hilbert's views as is evinced by these remarks Husserl made regarding axioms of completeness. A domain, Husserl told the Göttingen Mathematical Society, could conceivably be defined as complete or incomplete by axioms. Namely, if one has all the basic principles from which all possible propositions of the domain are derivable, then one has the complete theory of the domain. Formalizing these basic principles one obtains a formal deductive system in which each proposition has a corresponding formal proposition. Among the axioms already defining the domain, however, is an axiom of closure which stipulates that the domain is determined by certain axioms and no others. Where this axiom is not added the domain remains open insofar as further axioms can perhaps be added and the objects of the domain receive different formal interpretations. This, Husserl warned, is not legitimate completeness, not something specifically characteristic of axiom systems, because we can make any axiom system, any deductive system quasi complete by appealing to an axiom of that kind. So that kind of completeness can be of no use whatsoever to us. In extending an axiom system one obviously gives such an axiom up. A system of axioms with that kind of axiom cannot be extended. The concept of extending presupposes that no such axiom is involved. Moreover, Husserl continues, it is of course true that an axiom system closed from the outside in that spurious way already has the property sought, namely that one can tell whether or not a given proposition follows from the axioms or not. It need only contain the relations, forms of combination, in short the concepts, formally defined by the axiom system. If it has a meaning in terms of these definitions, it is either true in virtue of the definitions, or in contradiction with them (pp. 441-42).

Incidentally, Husserl considered the completeness of arithmetic to be self-evident because, as he explained to his Göttingen audience, any arithmetic, no matter whether it is defined to include just all positive numbers, all real numbers, positive rational numbers or rational numbers in general, etc., is defined by a system of axioms on the basis of which we can prove that any proposition derived exclusively from concepts established as being valid by the axioms either follows from the axioms or is in contradiction with them (p. 443).

4. Formal Logic and Complete Deductive Systems

Husserl's earliest struggles to provide the real number system with sound logical foundations and to establish the consistency of arithmetic soon evolved into a quest to secure sound logical foundations for all scientific knowledge. In 1929 he published Formal and Transcendental Logic (Husserl 1929) which was the product of decades of reflection upon the relationship between logic and mathematics, between mathematical logic and philosophical logic, between logic and psychology, and between psychologism and his own transcendental phenomenology. One of the stated goals of the book was to redraw the boundary line between logic and mathematics in light of the new investigations into the foundations of mathematics. A second goal was to examine the logical and epistemological issues such developments have raised (pp. 10-17, §11). The kinship with Hilbert's ideas is palpable, and the work surely benefited from years of direct participation in the events that had taken place in the mathematical world during his lifetime which would have had to condition any informed response to the questions Husserl was asking.

Now I want to look at some of the steps in Husserl's reasoning that led him from his questions about the consistency of the real number system to the actual development of a theory of formal logic with a theory of complete deductive systems as its highest task, and a transcendental logic as its complement. First, I need to look at how Husserl came to conceive the relationship between formal mathematics and formal logic.

Husserl believed that the formalization of large tracts of mathematics in the nineteenth century had laid bare the deep significant connections obtaining between formal mathematics and formal logic, and so had raised profound new questions about the deep underlying connections existing between the two fields. Logic and mathematics, he believed, had originally developed as separate fields because it had taken so long to elevate any particular branch of mathematics to the status of a purely formal discipline free of any reference to particular objects. Until that had been accomplished the important internal connections obtaining between logic and mathematics were destined to remain hidden. Once large tracts of mathematics had been formalized, however, parallels existing between its structures and those of logic became apparent, and the abstract, ideal, objective dimension of logic could be properly recognized, as it traditionally had been in mathematics.

In particular, developments in formalization had unmasked close relationships between the propositions of logic and number statements making it possible for logicians to develop a genuine logical calculus which would enable them to calculate with propositions in the way mathematicians did with numbers, quantities and the like (Husserl 1929, 23-27).

Extensionalist logicians, Husserl says, had also worked on the logical foundations of mathematics, and had come to some of the same conclusions he had. But in Formal and Transcendental Logic, Husserl condemns their work as fundamentally misguided and unclear. He qualifies extensional logic as naive, risky, and doubtful. He complains that it has been the source of many a contradiction requiring every kind of artful device to make it safe for use in mathematical reasoning. However, Husserl credits extensionalist logicians with having managed to raise some highly interesting philosophical questions, and with having succeeded in imparting a genuine sense of the common ground existing between mathematics and logic to mathematicians, whose work is relatively unhindered by the particular lack of clarity involved (Husserl 1929, 74, 76, 83).

Husserl turned to Bolzano for a theory of meaning and analyticity appropriate to the true logical calculus he now envisaged. Husserl's early work on the philosophy of arithmetic, I have argued, had convinced him that arithmetic was an analytic a priori discipline. Initially, Husserl had believed that Bolzano's theories regarding ideal propositions in themselves and truths in themselves involved an appeal to abstruse metaphysical entities, but in the 1890's it all of a sudden became clear to Husserl that Bolzano had actually been talking about something fundamentally completely understandable, namely the meaning of an assertion, what was declared to be one and the same thing when one says of different people that the affirm the same thing. This realization demystified meaning for Husserl.35

Husserl was persuaded of the inadequacy of Kant's analytic-synthetic distinction, and he came to believe that Bolzano's more Leibnizian approach to analyticity and meaning harbored the insights logicians needed to prove their propositions by purely logical means, meaning analytically in virtue of the meaning of their terms in a way analogous to mathematical reasoning. However, in Husserl's opinion, Bolzano never saw the internal equivalence between the analytic nature of both formal logic and formal mathematics made possible by developments in the field of mathematics that had only taken place after his death.36

At the same time, Husserl's familiarity with deductive techniques employed by his contemporaries drew him to see the advantages of deductive reasoning patterned after the formal reasoning advocated in formalist theories of mathematics. He believed this could be applied to formal logic as a technique for deriving propositions from propositions in a purely logical, analytic way. Hilbert's stringent requirements regarding consistency and completeness were much the same as Husserl's own ideas as to how propositions were to be derived from propositions.

Husserl believed the highest task of formal logic to be the theory of complete deductive systems and the complete manifolds which were their counterparts in the objective order. In Formal and Transcendental Logic §28, Husserl cites the characterization of complete deductive systems he had given in the 1900 Prolegomena §70 saying that he could not improve upon it. "The objective correlate of the concept of a possible theory determined only in its form", he had written there, "is the concept of any possible domain of knowledge that would be governed by a theory having such a form, what mathematicians call a manifold". It is then, according to his theory, a domain determined solely by the circumstance that it comes under a theory having such a form, that among the objects belonging to the domain certain connections are possible. In respect of their matter the objects remain completely indeterminate, are exclusively determined by the form of the combinations ascribed to them, combinations which are themselves only formally determined by the elementary laws assumed to hold good for them.

For Husserl, the great advance of mathematics as developed by Riemann and his successors consisted not just in appealing to the form of deductive systems, but also in having gone on to view such systems of forms as mathematical objects, to be altered freely, universalized mathematically.

Husserl wrote of what he called the hidden origin of the concept of a complete manifold that: "If the Euclidean ideal were actualized, then the whole infinite system of space-geometry could be derived from the irreducible finite system of axioms by purely syllogistic deduction according to the principles of lower level logic, and thus the a priori essence of space could become fully disclosed in a theory … the transition to form then yields the form idea of any manifold that conceived as subject to an axiom system by formalization could be completely explained nomologically in a deductive theory that would be 'equiform' with geometry." If, he continued, a manifold is conceived as defined and determined exclusively by such a system of forms of axioms belonging to the theorems and component theories, then ultimately the whole science forms necessarily valid for such a manifold can be derived by pure deduction.

And then Husserl asked just what it was that purely formally characterized a self-contained system of axioms as "complete", as a system by which actually a "manifold" would be defined. Every formally defined system of axioms, he noted, has an infinity of deducible consequences, but a manifold governable by an explanatory nomology includes the idea that there is no truth about such a domain that is not deducibly included in the "fundamental laws" of the corresponding nomological science; it is not defined by just any formal axiom system, but by a complete one. Such an axiom system is characterized by the fact "that any proposition … that can be constructed in accordance with the grammar of pure logic out of concepts … occurring in that system is either true, that is to say: an analytic (purely deducible) consequence of the axioms, --or false, --that is to say: an analytic contradiction--; tertium non datur." For Husserl such axiom systems represented the highest level of formal logic (Husserl 1929, 28-36).

5. Formal and Transcendental Logic

However, Husserl's logicians cannot leave matters there. For them formal logic alone cannot suffice in Husserl's sense of logic as a theory of science, an enlarged analytics. As theoreticians of science in general, they are also obliged to contend with the question of basic truths about a universe of objects existing outside of formal systems; they are called upon to seek solutions to the problems that come up when scientific discourse steps outside the purely formal domain and makes reference to specific objects or domains of objects. They are not free to sever their ties with nature and science, to accept a logic that tears itself entirely away from the idea of any possible application and becomes a mere ingenious playing with thoughts, or symbols that mere rules or conventions have invested with meaning. They must step out of the abstract world of pure analytic logic with its ideal, abstract entities, and confront those more tangible objects that make up the material world of things. In addition, they are obliged to step back and investigate the theory of formal languages and systems themselves, and their interpretations, to engage in what we call metatheory (Husserl 1929, 109-10, 52).

Even in playing a game one actually judges, collects, counts, and draws conclusions, Husserl points out. On the purely formal level complete manifolds could be viewed as deductive games with symbols. But one is not dealing with an actual theory of manifolds, he maintains, until one regards the game symbols as being signs for actual abstract objects, units, sets, manifolds, etc., and until the rules of the game acquire the status of laws applying to these manifolds (Husserl 1929, 99).

Husserl's logicians must also see that the logical sense of the formal sciences also includes a sphere of cognitive functioning and a sphere of possible applications, and they must also set about trying to answer the difficult questions regarding the way they themselves interact with both the objective structures of the abstract realm of formal logic and mathematics, and those of the material order. So once armed with the objective structures of formal logic, Husserl's logicians are still obliged to go further and to come to terms with really hard ontological questions about the objects involved and equally hard epistemological questions about subjectivity. Philosophical logicians cannot ignore these problems. Formal logic requires a complement in the form of what Husserl called a transcendental logic (Husserl 1929, 109, 111).

In the introduction to Formal and Transcendental Logic (Husserl wrote that the problem guiding him originally lay in determining the sense of, and isolating, a pure analytic logic of non-contradiction was one of evidence: the problem of the evidence of the sciences making up formal mathematics. He was, he says, particularly struck by the fact that the evidence making up the truths of formal mathematics and formal logic is of an entirely different order than that of other a priori truths in that the former do not involve any intuition of objects or states of affairs whatsoever (Husserl 1929, 12). The formalness of these disciplines lies precisely in their relationship to "anything whatever", with a most empty universality which makes no reference to any actual material interpretation, to any material particularly characterizing the objects or domain of objects. (Husserl 1929, 87)

Mathematics has its own purity and legitimacy. Mathematicians are free to create arbitrary structures. They need not be concerned with questions regarding the actual existence of their formal constructs, nor with any application or relationship their constructs might have to possible experience, or to any transcendent reality. They are free to do ingenious things with thoughts or symbols that receive their meaning merely from the way in which they are combined, to pursue the necessary consequences of arbitrary axioms about meaningless things, restricted only by the need to be non-contradictory and coordinated to concepts previously introduced by precise definition. (Husserl 1929, 138)

And the same, Husserl contends, is true for formal logic when it is actually developed with the radical purity which is necessary for its philosophical usefulness and gives it the highest philosophical importance. Severed from the physical world, it lacks everything that makes possible a differentiation of truths or, correlatively of evidences. However, real philosophical logic, in Husserl's sense of a theory of science, a Wissenschaftslehre, can only develop in connection with transcendental phenomenology by which logicians penetrate an objective realm which is entirely different from them. (Husserl 1929, p. 8, §23)

Husserl's logicians are not brain dead machines. Far from it. His logicians can only submit to a logic which they have thought through and thought through with insight, a fact which he believed cried out for epistemological investigations into the subjective and intersubjective cognitive processes that inevitably interact with the objective order.

For Husserl, as for Bertrand Russell and Jean Paul Sartre, it was Brentano's theory of the intentionality of mental acts that indicated the way out of the mind to things. According to Brentano's famous thesis of intentionality every mental phenomenon is characterized, by reference to a content, direction toward an object. For Husserl intentionality acted as the philosophical logician's bridge from the mind to the objective order, and was the key that unlocked the way to transcendental analyses which chart the mind's path to things (Husserl 1929, §§ 42-44, 210, 245). Husserl was keenly aware of the pitfalls of subjectivity and as a part of his project to overcome them he undertook exhaustive studies of the knowing process, finally prescribing a demanding series of mental exercises designed to instill rigor into epistemology by training philosophers to reason (Husserl 1929, 274) in ways which Husserl hoped would make the psychological chaff fly, and the transcendental grain lie sheer and clear (Husserl 1929, 13, 237, §61).

Hilbert too believed that his formal logic alone could not suffice "If scientific knowledge is possible," he maintained, "certain intuitive conceptions and insights are indispensable."37 "No more than any other science", he wrote,

can mathematics be founded by logic alone; rather as a condition for the use of logical inferences and the performance of logical operations something must already be given in our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else not requires reduction. This is the basic philosophic position that I regard as requisite for mathematics and, in general, for all scientific thinking, understanding and communication.38

In many respects Husserl may be viewed as one who endeavored to provide the philosophical complement to Hilbert's views.

Before concluding, I would like to draw attention to an additional difference between Husserl and Frege. In Formal and Transcendental Logic, Husserl also uses the example of identity statements to point out differences between the formal order and the material order which are relevant to philosophers who have followed the discussions on sense and reference in this century. This is in reference to identity statements of the form "a is b", and "a is c". Husserl argues that the objects designated in these statements are non-self-sufficient under all circumstances. They are what they are within the context of the whole, and different wholes can have components which are equal, but not the same in all ways. If we say "a is b" and "a is c", the a in the first statement is not identical to the a of the second, he maintains. The same object a is meant in both cases, but in a different how (Husserl 1929, 295-96).

Mathematicians, Husserl acknowledges, are not in the least interested in the different ways objects may be given. For them objects are the same which have been correlated together in some self-evident manner. However, Husserl warns, logicians who do not bewail the lack of clarity involved here, or who claim that the differences do not matter are not philosophers, since here it is a matter of insights into the fundamental nature of formal logic, and without a clear grasp of the fundamental nature of formal logic, one is obviously cut off from the great questions that must bc asked about logic and its role in philosophy (Husserl 1929, 147- 48).

Here we have a fundamental and abiding difference between Husserl and Frege who always insisted that in his logic, as in mathematics, there was no difference between identity and equality.39 Husserl had taken issue with Frege on this very matter in the Philosophy of Arithmetic,40 and as can be imagined from what has just been said, Husserl's own views on the question played a determinant role in all his philosophical and logical investigations.

6. Conclusions

The formalistically inclined thinker that I have described as hard at work rigorizing philosophical thought is a far cry from being the man whom both detractors and disciples have so often depicted as wantonly engaging in an orgy of subjectivity. Husserl did have a distressing propensity to deal with everything that had to do with subjectivity, so those portrayals are not utterly without foundation. But he did insist that the subjective order could not be properly examined until the objective order had been, and until the objectivity of the structures girding scientific knowledge had been established and demonstrated. He maintained that pure logic with its abstract ideal structures would have to be clearly seen and definitely apprehended as dealing with ideal objects before transcendental questions about them could be asked (Husserl 1929, §§ 8, 9, 11, 26, 42-44, and pp. 81-82, 111, 225, 246, 258, 263, 266).

In the beginning of this paper I wanted to inquire into the deep, underlying reasons that may have drawn Husserl and Hilbert to want to establish a criterion of completeness for formal reasoning. If any definitive answer to my question is possible, finding it would require a vastly more thoroughgoing investigation than is possible here. In particular, a great deal more would have to be known about Husserl's encounter with what Hilbert called "Cantor's majestic world of ideas".4l I hope, however, that I have at least succeeded in showing that Husserl's and Hilbert's reasons were tied in with their conviction that thc real number system could only be grounded by a logic that countenanced reference to ideal entities. A logic that could not cope with expressions whose, in Husserl's words, "absurdity is mediate, i.e. the countless expressions shown by mathematicians in lengthy indirect demonstrations, to be objectless a priori" could never, they thought, provide secure foundations for knowledge.

I would also like to suggest that approaching Husserl's thought in light of his views on completeness, analyticity, meaning and identity may also help demystify his phenomenology and so shed light and order where confusion and ineffability have seemed to many to reign. And it could provide keys to understanding some things that have seemed inaccessible about Husserl's thought to those schooled in the logical and epistemological views fashioned by philosophers for whom a rival philosophy of logic and mathematics has occupied a central position in their philosophical views. Husserl was far from ignorant of the developments in mathematics and logic that made Principia Mathematica and related systems possible, nor did he turn his back on those developments. He worked long and hard to resolve questions raised by them that are still under discussion today and his exhaustive studies merit study now.

Undertaking such a job, however, is not for the fainthearted for they will find themselves investigating the workings of the human mind, and trafficking in intensional phenomena like concepts, essences, properties and attributes, not to mention courting the a priori and the ideal. Husserl devoted his life to investigating these irksome phenomena which many others have hoped dearly to eradicate by rigorously applying techniques inspired by another logic. In his attempts to find clarity with respect to the central traits of reality, Husserl in fact incorporated into his philosophy almost everything extensionally minded philosophers hoped to ban.

* This paper is based on a paper read in April 1992 at a conference on the development of the foundations of mathematics from 1850-1930 organized by Professor Jaakko Hintikka, which was part of the program of the Boston Colloquium for the Philosophy and History of Science. I must thank Mr. Dagfinn Føllesdal, who also participated in the session, for his thorough reading of my text and his insightful suggestions as to how to improve it.

This paper was, to my knowledge, the first article length work to have been written on the subject. Researching it persuaded me of the need to investigate the relationship of Husserl's ideas to those of Georg Cantor before coming to any further conclusions about the relationship between Husserl's and Hilbert's ideas. Chapters 6, 7 and 8 of this book are the fruit of those efforts.

Chapter 9, written in 1998 and 1999, benefited from that subsequent research on Cantor and Husserl and also from the publication of Husserl's Logik und allgemeine Erkenntnistheorie for the first time in 1996, which has an entire chapter on Husserl's theory of manifolds that is incomparably clearer and more explicit than what is to be found in his other available writings.

In addition, I have since had many exchanges with Professor Jairo da Silva of the Mathematics Department of the University of São Paulo. He has recently convinced me of the importance of the distinction Husserl makes between relative completeness and the absolute completeness of Hilbert. Look for his paper "Husserl's Two Notions of Completeness", forthcoming in Synthese. Much research remains to be done on the relationship between Husserl's and Hilbert's ideas.

Notes

1D. Hilbert's "Über den Zahlbegriff" was first published in the Jahresbericht der Deutschen Mathematiker-Vereinigung 8, 1900, pp. 180-84, and subsequently as an appendix to post-1903 editions of his Grundlagen der Geometrie. I have cited the translation of Hilbert's axioms for arithmetic appearing in M. Kline, Mathematical Thought From Ancient to Modern Times, Vol. 3, Oxford University Press, New York, 1972, pp. 990-91.

2 See E. Husserl's Ideas, General Introduction to Pure Phenomenology, Colliers, New York, 1962 (1913), § 72 and note, his Formal and Transcendental Logic, M. Nijhoff, The Hague, 1969 (1929), §§ 28-36, and his The Crisis of European Sciences and Transcendental Phenomenology, Northwestern University Press, Evanston, 1970, § 9f and note. In these texts Husserl refers back to his discussions in the Logical Investigations, Humanities Press, New York, 1970, Prolegomena, §§ 69 and 70 and to the then unpublished material from his Göttingen period, now published in appendices to his Philosophie der Arithmetik, mit ergänzenden Texten, Husserliana Vol. XII, M. Nijhoff, The Hague, 1970. Of particular interest is the chapter on manifolds in the recently published Logik und allgemeine Wissenschaftstheorie, Husserliana Vol. XXX, Kluwer, Dordrecht, 1996.

As usual there are some terminological obstacles that make it hard to see the connection Husserl's ideas have with the logical tradition most familiar to readers of English. First of all, for complete and completeness Husserl uses the German words 'definit' and 'definitheit' in the place of Hilbert's 'vollständig' and 'vollständigkeit'. Since in the passages cited above Husserl maintains that his concept of definitheit is exactly the same as Hilbert's vollständigkeit, I have tried to avoid the terminological confusion by translating Husserl's terms by the more familiar 'complete' and 'completeness', although Husserl translators have understandably chosen 'definite' and 'definiteness'. Second, in the above texts Husserl refers to his theory of complete Mannigfaltigkeiten, a term which has been translated by 'multiplicity' or 'manifold'. For Husserl complete Mannigfaltigkeiten are the objective correlates of complete axiom systems.

3 S. Bachelard, A Study of Husserl's Formal and Transcendental Logic, Northwestern Press, Evanston, 1968, pp. 59-61; J. Cavaillès, Sur la logique et la théorie de science, Presses Universitaires de France, Paris, 1947, pp. 70, 73; R. Schmit, Husserls Philosophie der Mathematik, Bouvier, Bonn, 1981, pp. 67-86. H. Lohmar, Husserls Phänomenologie als Philosophie der Mathematik, Dissertation, Cologne, 1987, pp. 151-62; Guillermo E. Rosado Haddock, Edmund Husserls Philosophie der Logik und Mathematik im Lichte der gegenwärtigen Logik und Grundlagenforschung, Dissertation, Rheinischen Friedrich-Wilhelms Universität, Bonn, 1973; B. Picker, "Die Bedeutung der Mathematik für die Philosophie Edmund Husserls", Philosophia Naturalis 7, 1962, pp. 266-355, his Dissertation, Münster, 1955.

4 See for example the note to Husserl's Ideas § 72.

5 K. Schuhmann, Husserl-Chronik, M. Nijhoff, The Hague, 1977, pp. 6-11. I also discuss Husserl's background throughout my Word and Object in Husserl, Frege and Russell: Roots of Twentieth Century Philosophy, Ohio University Press, Athens OH, 1991 and in "Husserl and Frege on Substitutivity", Chapter 1 of this book.

6 L. McAlister, The Philosophy of Franz Brentano, Duckworth, London, 1976, pp. 45, 49, 53; A. Osborn, The Philosophy of E. Husserl in its Development to his First Conception of Phenomenology in the Logical Investigations, International Press, New York, 1934, pp. 12, 17, 18, 21; M. Dummett, The Interpretation of Frege's Philosophy, Harvard University Press, Cambridge, MA, 1981, pp. 72-73, and his Frege, Philosophy of Language, Duckworth, London, 2nd ed. rev., 1981, p. 683; Hill, Word and Object in Husserl Frege and Russell, pp. 59-67 and Chapter 7.

7 A. Fraenkel, "Georg Cantor", Jahresbericht der Deutschen Mathematiker Vereinigung 39, 1930, pp. 221, 253 n., 257; E. Husserl, Introduction to the Logical Investigations, M. Nijhoff, The Hague, 1975, p. 37 and notes; J. Cavaillès, Philosophie Mathématique, Hermann, Paris, 1962, p. 182; Schmit, pp. 40-48, 58-62; L. Eley, "Einleitung der Herausgebers" to Husserl's Philosophie der Arithmetik, mit ergänzenden Texten, pp. XXIII-XXV; Georg Cantor Briefe, H. Meschkowski and W. Nilson (eds.), Springer, New York, 1991, pp. 321, 373-74, 379-80, 423-24. Two Cantor letters dating from 1895 are published in W. Purkert and H. Ilgauds, Georg Cantor 1845-1918, Birkhäuser, Basel, 1991, pp. 206-07.

8 Kline, Mathematical Thought From Ancient to Modern Times, Vol. 3, pp. 950-56, 960-66; McAlister, The Philosophy of Franz Brentano p. 49; Osborn, The Philosophy of E. Husserl in its Development to his First Conception of Phenomenology in the Logical Investigations p.18; Husserl, Introduction to the Logical Investigations, p. 37.

9 H. Wang, From Mathematics to Philosophy, Routledge and Kegan Paul, London, 1974 pp. 145-52 in reference to B. Bolzano's 1837 Wissenschaftslehre §§ 148 and 155. Bolzano's book has been partially translated as Theory of Science by R. George, Oxford: Blackwell, 1972 and B. Terrell, Reidel, Dordrecht, 1973.

10 See the Husserl texts cited in note 1.

11 A. Whitehead, An Introduction to Mathematics, Oxford University, Oxford,

Press, 1958 (1911), pp. 62-64

12 Appendix Vl to Husserl's Philosophie der Arithmetik mit ergänzenden Texten, p. 433 and his Formal and Transcendental Logic, § 31.

13 Husserl, Introduction to the Logical Investigations, pp. 33-36.

14 Hill, Word and Object in Husserl, Frege and Russell, pp. 80-95

15Good accounts of Husserl s work during the 1890s are given in the editors' introductions to the Husserliana editions of Husserl's Philosophie der Arithmetik mit ergänzenden Texten (Vol. XII), Logische Untersuchungen (Vol. XVIII) and Studien zur Arithmetik und Geometrie (Vol. XXI).

16 E. Husserl, Philosophie der Arithmetik, Pfeffer, Halle, 1891, p. viii (note this

is not the Husserliana edition cited above for the posthumously published material, but

Husserl's 1891 book). Hill, Word and Object in Husserl, Frege and Russell, pp. 84-86

17 Husserl, Philosophie der Arithmetik, mit ergänzenden Texten, pp. 340-429

18 Cited in Hill, Word and Object in Husserl, Frege and Russell, p. 85. See also D. Willard, Logic and the Objectivity of Knowledge, Ohio University Press, Athens OH, 1984, pp. 115-16.

19 Husserl, Philosophie der Arithmetik (1891), pp. 104-05 132-34. I discuss his arguments in depth in "Husserl and Frege on Substitutivity", Chapter 1 of this book.

20 Husserl, Logical Investigations, Prolegomena p. 179 n. Husserl actually retracted pp. 129-32, not pp. 124-32 as a typographical error in the English edition indicates.

21 Husserl, Philosophie der Arithmetik (1891) pp. 130-31

22 G. Frege, Philosophical and Mathematical Correspondence, Blackwell, Oxford, 1980, p. 65, in reference to Frege's article "On Formal Theories of Arithmetic", Collected Papers on Mathematics, Logic and Philosophy, B. McGuinness (ed.), Oxford, Blackwell, 1984, pp. 112-21.

23 Frege's Collected Papers, pp. 118-19. Husserl's own copy of Frege's article is now in the Husserl library at the Husserl Archives in Leuven, Belgium.

24 G. Frege, Posthumous Writings, Oxford, Blackwell, 1979, p. 122. See also G. Frege, Translations from the Philosophical Writings, Oxford, Blackwell, 3rd ed.,1980, pp. 22-23, 32-33, 162-213.

25 Frege, Translations from the Philosophical Writings, pp. 69-70.

26 Frege, Posthumous Writings, p. 123

27 D. Hilbert, "On the Infinite", From Frege to Gödel, J. van Heijenoort (ed.), Harvard University Press, Cambridge, MA, 1967, p. 379

28 Ibid., p. 383

29 See the introduction to Husserl's Logical Investigations, Husserliana Vol. XVIII.

30 See the introduction to Studien zur Arithmetik und Geometrie, Husserliana Vol. XXI, M. Nijhoff, The Hague, 1984, p. XII, where a 1901 letter from Husserl's wife is cited.

31 C. Reid's Hilbert, Springer, New York, 1970, pp. 67-68 and Hilbert and Courant in Göttingen and New York, Springer, New York, 1976 provide anecdotal material about Husserl's time in Göttingen; Schuhmann, Husserl-Chronik, p. 10, Husserl's thesis entitled Beitrag zur Theorie der Variationsrechnung

32 As Hilbert makes evident in "On the Infinite", Frege to Gödel, Van Heijenoort (ed.),

pp. 369-92.

33 Frege, Philosophical and Mathematical Correspondence, pp. 34-51, and Gottlob Freges Briefwechsel mit D. Hilbert, E. Husserl, B. Russell, Meiner, Hamburg, 1980, pp. 3, 47. Also my "Frege's Letters", From Frege to Dedekind, J. Hintikka (ed.), Kluwer, Dordrecht, 1995, pp. 97-118.

34 The notes for Husserl's lecture are published as an appendix to Philosophie der Arithmetik, mit ergänzenden Texten, pp. 430-506. They are cited in the text as (Husserl 1901). Concerning the invitation see Husserl's wife's letter cited in note 30.

35 E. Husserl, "Review of Melchior Palágyi's Der Streit der Psychologisten und Formalisten in der modernen Logik ", Early Writings in the Philosophy of Logic and Mathematics, Kluwer, Dordrecht, 1994, p. 201.

36 Husserl, Introduction to the Logical Investigations, pp. 36-38, 48. Formal and Transcendental Logic, pp. 184-85, 225.

37 Van Heijenoort (ed.), p. 392.

38 Ibid., pp. 464-65, and 376.

39 Frege, Translations From the Philosophical Writings, pp. 22-23, 120-21, 141n., 146n., 159-61, for example.

40 I discuss this at length in my Word and Object in Husserl Frege and Russell, especially Chapter 4, and in "Husserl and Frege on Substitutivity", Chapter 1 of this book.

41 From Frege to Gödel, Van Heijenoort (ed.), p. 437. Since I wrote this I myself have investigated the relationship between Husserl's and Cantor's ideas. See Chapters 6, 7, 8 and 9 of this book.

42 Husserl, Logical Investigations, First Investigation, pp. 293-94.