This is a preprint version of the paper which appeared in definitive form in Axiomathes, an International Journal in Ontology and Cognitive Systems, vol. 13, no. 1, 2002, pp. 79-104. The published version should be consulted for all citations.

Tackling Three of Frege's Problems: Edmund Husserl on Sets and Manifolds

by Claire Ortiz Hill

Abstract. Edmund Husserl was one of the very first to experience the direct impact of challenging problems in set theory and his phenomenology first began to take shape while he was struggling to solve such problems. Here I study three difficulties associated with Frege's use of sets that Husserl explicitly addressed: reference to non-existent, impossible, imaginary objects; the introduction of extensions; and "Russell's" paradox. I do so within the context of Husserl's struggle to overcome the shortcomings of set theory and to develop his own theory of manifolds. I define certain issues involved and discuss how Husserl's theory of manifolds might confront them. In so doing I hope to help bring Husserl's theories about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of uncharted philosophical territory rich in philosophical implications.

1. Introduction

Gottlob Frege was the first to admit that the sets the way he introduced in the Foundations of Arithmetic (Frege 1884) and endeavored to legitimate them in Basic Laws of Arithmetic (Frege 1893) was problematical. On several occasions, he admitted that it was only with difficulty that he had done so because the matter had not seemed to him to be quite secure. He even alluded to having been in a certain way forced to introduce them in the way he did almost against his will because he saw no other alternative.

In this essay, I study three problems associated with Frege's use of sets: 1) that of reference to non-existent, impossible, imaginary objects; 2) the introduction of extensions; and 3) Frege's true bête noire, the paradox publicized by Bertrand Russell. I address these three problems within a discussion of the struggle to overcome the shortcomings of set theory engaged in by Edmund Husserl, a philosopher not generally perceived, even by Husserl scholars who should know better, as having been conversant with set theory at all. In so doing, I hope to stimulate investigation into important theories of Husserl's that stand to have interesting applications to philosophy nowadays

Indeed, although the fact that investigations into set theory and the philosophy of arithmetic by mathematicians turned philosophers played a pre-eminent role in the shaping of 20th century western philosophy is perfectly obvious to anyone familiar with the origins of analytic philosophy, the same cannot be said of philosophers' perceptions of the origins of the phenomenological school. However, Husserl was one of the very first to experience the direct impact of challenging problems with set theory and his phenomenology first began to take shape while he was struggling to solve such problems (Hill and Rosado Haddock 2000). First, however, something needs to be said about some terminological obstacles.

2. Terminological Considerations

Terminologically, it is absolutely necessary to remember that Husserl's theories about sets and manifolds developed during the fifteen years that he spent in the company of Georg Cantor, the creator of set theory and the author of the Mannigfaltigkeitslehre. So use of the term 'Mannigfaltigkeit', usually translated into English by 'manifold' or 'multiplicity', cannot be univocal within the context of the problems addressed in this essay.

Cantor used the terms 'Menge', 'Mannigfaltigkeit' and 'Inbegriff' interchangeably (Grattan-Guinness 2000, Chapter 3). Right from the start Husserl, however, directly confronted the terminological difficulties that one faces in speaking of Mannigfaltigkeiten. In his earliest writings, he noted that in place of the word 'Vielheit' the practically synonymous terms 'Mehrheit', 'Inbegriff', 'Aggregat', 'Sammlung', 'Menge', etc. (variously translated by 'quantity', 'aggregate', 'plurality', 'totality', 'collection', 'set', 'multiplicity') had been used and he acknowledged the ambiguity thus introduced into reasoning. In the beginning of Philosophy of Arithmetic, he informed readers that, while recognizing the differences, he would not initially restrict himself to using any one of these terms exclusively. He said that he hoped thereby to be able to neutralize differences in meaning among them (Husserl 1891, 8, note). This same strategy had been adopted in On the Concept of Number (Husserl 1887, 96).

Although Husserl did use the various terms for set interchangeably in the late 1880s, in those days he rarely used the more Cantorian term 'Mannigfaltigkeit', which he only began to use more frequently in posthumous writings of the 1890s and, when he particularly studied geometrical manifolds (see, for example Husserl 1983). By that time, the manifolds of the Mannigfaltigkeitslehre that Husserl himself was developing, and that he ultimately considered to be the highest expression of pure logic, were quite different from Cantor's Mannigfaltigkeiten (Hill and Rosado Haddock 2000, Chapters 7,8,9). From that time on, he strove to distinguish sets from manifolds, multiplicities, totalities, aggregates, and so on.

In what follows I translate 'Vielheit' as 'multiplicity'; 'Allheit' as 'totality'; 'Inbegriff' as 'aggregate'; 'Menge' as 'set'; 'Mannigfaltigkeit' as 'manifold'. I translate 'Widerspruch' by 'contradiction' and 'Widersinnigkeit' by contradiction in terms. The translations appearing in the text are often my own and I have often modified published translations to make the terminology consistent.

3. Sets in On the Concept of Number and Philosophy of Arithmetic

In the Philosophy of Arithmetic, Husserl endeavored to instill rigor into the rather inchoate set theoretical foundations being proposed for numbers (Hill and Rosado Haddock 2000, Chapters 7, 8). In Formal and Transcendental Logic, he even explicitly characterized the Philosophy of Arithmetic as having represented an initial attempt on his part "to obtain clarity regarding the original genuine meaning of the fundamental concepts of the theory of sets and cardinal numbers" ("Klarheit über den ursprungssechten Sinn der Grundbegriffe der Mengen- und Anzahlenlehre zu gewinnen") (Husserl 1929, §27a; also §24 and note).

In both On the Concept of Number and the Philosophy of Arithmetic, Husserl in fact tried to make set theory the basis of arithmetic. Just as cardinal numbers relate to sets (Mengen), so ordinals relate to series which are themselves ordered sets (Mengen), he maintained in the introduction to Philosophy of Arithmetic (Husserl 1891, 4), chapter 1 of which opens with the assertion that the "analysis of the concept of number presupposes that of the concept of multiplicity (Vielheit)". There cannot be any doubt, Husserl went on to affirm there, as in On the Concept of Number, that the concrete phenomena that form the basis for the abstraction of the concepts in question are aggregates (Inbegriffe), multiplicities (Vielheiten) of determinate objects. Everyone, he stressed, knew what was meant by this, for despite difficulties experienced in analyzing it, the concept of Vielheit itself was perfectly precise and the range of its extension exactly delimited. It might therefore be considered as a given, he maintained, even though we might still be in the dark about the essence and formation of the concept itself (Husserl 1891, 9-10, 13; Husserl 1887, 96-97, 111).

Husserl then turned to study the experience of concrete Inbegriffe in order to consider how both the more indeterminate universal concept of Vielheit and the determinate number concepts were to be abstracted from them. Crucial to that account was his conviction that any talk of such Inbegriffe necessarily involved "collective combination", the combination of the individual contents into a whole, into a unity containing the individual contents as parts. Though the combination involved might be very loose, there was, the faithful student of Brentano held, a particular sort of unification there that would also have to have been noticed as such since the concept of Inbegriffs (der Vielheit) could never have arisen otherwise. The concept of Vielheit, he explained, in fact arose through reflection upon the special way in which the contents were united together in each concrete Inbegriff, a way analogous to the way in which the concept of any other kind of whole arose through reflection upon the mode of combination peculiar to it (Husserl 1891, 13-15; Husserl 1887, 97-98).

In On the Concept of Number, Husserl characterized the distinctive abstractive process that yielded the concept of set in this way:

It is easy to characterize the abstraction which must be exercised upon a concretely given Vielheit in order to attain to the number concepts under which it falls. One considers each of the particular objects merely insofar as it is a something or a one herewith fixing the collective combination; and in this manner there is obtained the corresponding general Vielheitsform, one and one and . . . and one, with which a number name is associated. In this process there is total abstraction from the specific characteristics of the particular objects.... To abstract from something merely means to pay no special attention to it. Thus in our case at hand, no special interest is directed upon the particularities of content in the separate individuals (Husserl l887, ll6-17).

In the Philosophy of Arithmetic Husserl studied this same process in greater detail (in Chapter 4 especially), especially underscoring the uniqueness of the abstraction process yielding the number concept. There, number was characterized as the general form of a Vielheit under which the Inbegriff of objects a, b, c fell. "It is thus clear", Husserl wrote, "that this Inbegriff (this Vielheit, Menge, or whatever one may want to call it) makes up the subject of the number statement. Considered formally, the number and the concrete set (Menge) are related to one another as concept and object. The number is not consequently assigned to the concept of the objects counted, but to their Inbegriff" (Husserl 1891, 185-66).

In Chapter XI of Philosophy of Arithmetic, Husserl took up what he called a particularly remarkable way of extending the original concept of Menge or Vielheit that by its very nature reached beyond the necessary bounds of human cognition, thus winning for itself an essentially new content. Signs, symbolic presentation à la Brentano, he explained, might aid the mind in reasoning in regions of thought beyond what could be known through direct cognitive processes like perception or intuition; the repeated application of operations permitting the collecting together of a multitude of objects one after the other into a set could take the place of the direct cognitive grasp of Mengen with hundreds, thousands or millions of members, and this was a way of actually representing collections in an ideal sense and essentially unproblematic from a logical point of view.

The logical problems connected with infinite Inbegriffen, Mengen, Vielheiten were, however, he maintained, of a completely different order, since the very principle by which they were formed or symbolized itself immediately made collecting all of their members together, one by one, a logical impossibility. By no extension of our cognitive faculties could one conceivably cognitively grasp or even successively collect such sets. With them one had reached the limits of idealization. As examples of infinite sets, he proposed the extensions of most general concepts, the set of the numbers of the symbolically extended number series, the set of points on a line and of the boundaries of a continuum (Husserl 1891, 24--6-48).

With infinite Menge, Husserl explained, one was in possession of a clear principle by which to transform any already formed concept of a certain given species into a new concept plainly distinct from the previous one. And this could be done over and over in such a way as to be certain a priori of never coming back to the original concept and to previously generated concepts. Repeated application of this process would yield successive presentations of continually expanding sets, and if the generating principle was really determinate, then it was determined a priori whether or not any given object could belong to the concept of the expanding set of concepts. In the case of the concept of the infinite Menge of numbers, one began with a direct presentation. The other natural numbers could be reached by repeated additions of 1, and nothing prevented one from advancing indefinitely in this way to new cardinals; we have a method for adding one to a previously given number, an operation necessarily generating one new number after another without return and limitation, each new number being determined by the process (Ibid.).

It is easy, Husserl added, to see why mathematicians had tried to transpose the concept of quantity onto such constructions, which were, however, of an essentially different logical nature. In the usual cases, the process by which the sets were generated was finite, there was always a last stage, and it was sometimes possible actually to bring the process to a halt and also to construct the corresponding set. But, this was quite absurd in the case of infinite sets. The process used to generate them was non-terminating, and the idea of a last stage, of a last member of the set meaningless. And this constituted an essential logical difference (Ibid.)

However, Husserl saw, despite the absurdity of the idea, analogies fostered a tendency to transpose the idea of constructing a corresponding collection for infinite sets, thereby creating what he called a kind of "imaginary" concept whose anti-logical nature was harmless in everyday contexts precisely because its inherent contradictoriness was never obvious in life. This was, Husserl explained, the case when "All S" was treated as a closed set. However, he warned, the situation changes when this imaginary construct was actually carried over into reasoning and influenced judgments. It was clear, he concluded, that from a strictly logical point of view we must not ascribe anything more to the concept of infinite sets than is actually logically permissible, and above all not the absurd idea of constructing the actual set (Ibid.).

4. Husserl's Growing Doubts About Sets

Husserl quickly came to judge his first attempts to clarify the true meaning of the fundamental concepts of the theory of sets and cardinal numbers to have been a failure. While working on the logic of mathematical thought and mathematical calculation, he explained in introductions to the Logical Investigations, he had encountered disturbing problems while studying the logic of formal arithmetic and the theory of Mannigfaltigkeiten. The first volume of the Logical Investigations, he explained in the foreword to the first edition, had arisen out of reflections upon certain implacable problems that had constantly hindered, and ultimately interrupted, the progress of many years of work devoted to achieving a philosophical clarification of pure mathematics. In a later preface to the book, he confessed to having been "unsettled --even tormented" by doubts about the psychological analysis of Mengen from the very start (Husserl 1900-01, 41-43; Husserl 1975, 16-17).

There was certainly some truth, Husserl concluded, in the idea that the idea of Menge arose out of collective combination. The Kollektivum was no material unity forming in the content of the items collected, he had initially reasoned, so the concept of collection had to arise through psychological reflection, in Brentano's sense, upon the act of collecting. But Husserl came to believe that the concept of number then had to be something basically different from the concept of collecting, which was all that could result through reflection on acts (Husserl 1975, 34-35). Such doubts eventually undermined his confidence in Brentano's theories, as well as those of Weierstrass and Cantor (Hill and Rosado Haddock 2000).

Husserl's antipathy towards a calculus of classes is evident in articles published during the 1890s (Husserl 1994, 92-114, 115-30, 135-38, 199, 443-51), where he sought to show "that the total formal basis upon which the class calculus rests is valid for the relationships between conceptual objects," and that one could solve logical problems without "the detour through classes", which he considered to be "totally superfluous" (Husserl 1994, 109, 123).

Husserl's chief target in those articles was Ernst Schröder, whom he credited with having provided "the best present form of the extensional calculus" (Husserl 1994, 52-91, 123, 421-41). Of particular interest here are Husserl's comments on Schröder's attempt to show that bringing all possible objects of thought into a class gives rise to contradictions. Schröder's argument, Husserl wrote in an article of 1891, might appear astonishing at first glance, but was actually sophistical. Husserl was, though, tellingly prepared to concede that:

in the case where we simultaneously have, besides certain classes, also classes of those classes, the calculus may not be blindly applied. In the sense of the calculus of sets as such, any set ceases to have the status of a set as soon as it is considered as an element of another set; and this latter in turn has the status of a set only in relation to its primary and authentic elements, but not in relation to whatever elements of those elements there may be. If one does not keep this in mind, then actual errors in inference can arise (Husserl 1994, 84-85).

In Halle, Husserl had been sort of a victim avant la lettre of the crisis in foundations that broke out once Russell began advertising the famous contradiction about the set of all sets that are not members of themselves that he discovered while studying Cantor's theories. Once appointed to Göttingen in 1901, Husserl was drawn into mathematical discussions with mathematicians who had known the contradiction for some years before Russell publicized his finding. However, no account of Husserl's reaction to the set theoretical paradoxes has yet to be fully pieced together, and it is only in unpublished notes on set theory (Husserl Ms A 1 35) , that we find him directly grappling with the questions raised by them. Though the ideas expressed there are not developed into a systematic whole, they yield clues into his views on matters not explicitly addressed in his published writings.

5. Husserl Develops His Own Mannigfaltigkeitslehre

Even as he was cogitating on set theory, whether Cantor's, Frege's, or Schröder's, Husserl was at work developing his own theory of Mannigfaltigkeiten. Between 1886 and 1893, he related in a 1901 letter, he had busied himself with the theory of geometry, formal arithmetic and the theory of manifolds, at times exclusively devoting himself to this (Husserl 1983, 396). In prefaces to the Logical Investigations, he specifically alluded to having been troubled by the Mannigfaltigkeitslehre (Husserl 1900-01, 41; Husserl 1975, 35).

Of the 1890s, Husserl wrote that he had seen about him only ambiguously defined problems and profoundly unclear theories, that disappointed by the theories of those to whom he owed most of his intellectual training and sick of all the confusion, he had abandoned work on the second volume of the Philosophy of Arithmetic about the logic of the deductive sciences and set out in search of his own answers, engaging in very general reflections that took him beyond the confines of the mathematical realm towards a universal theory of formal deductive systems. Traditional logic, he complained, should have made the rational nature of the deductive sciences, their formal unity and symbolic method, transparent and easily understandable, but the study of the actual deductive sciences left all that problematic and obscure. Husserl saw the mathematical theory of manifolds as a realization of the idea of a science of possible deductive systems, but viewed it, and all of modern formal analysis as but a partial realization of his own ideal of a science of possible deductive systems (Husserl 1900-01, 41-42; Prolegomena, § 70; Husserl 1975, 16-17, 35; Husserl 1994, 490-91).

In notes on sets and manifolds dating from 1891/92, one finds Husserl asking what in definition of the concept of Menge, of Mannigfaltigkeit in the broadest sense of the word, leads to the concept of Mannigfaltigkeit in the narrower sense. He begins to answer his question by noting that by Mannigfaltigkeit Cantor merely meant an aggregate of any elements combined into a whole and cites the passage of the Mannigfaltigkeitslehre where Cantor wrote: "By Mannigfaltigkeit or Menge, I understand generally any many that can be thought of as one, i.e., any aggregate of determinate elements which can be bound together into a whole by a law…." (Husserl 1983, 95)

Husserl goes on to note that Cantor's concept does not correspond to Riemann's and other related ones in the theory of geometry. For these, Husserl stresses, a Mannigfaltigkeit is an aggregate of elements that are not just combined into a whole, but are ordered and continuously interdependent (Husserl 1983, 95-96). Husserl defined order as "a concatenation that has the special property that each member possesses an unambiguous position in the narrow sense of the word in relation to any arbitrary one, i.e., can therefore be unequivocally characterized by the mere form of the direct or indirect connection with the last one" (Husserl 1983, 93). A Mannigfaltigkeit is not an aggregate of elements without relations, he underscored in 1892. It is precisely the relations that are essential and serve to distinguish it from a mere aggregate (Husserl 1983, 410).

In the Prolegomena to a Pure Logic, after his decade of lonely, intellectual struggle, Husserl unveiled his own theory of manifolds as the highest task of formal logic (Husserl 1900-01, Prolegomena, §§ 69-70). It is also discussed in Ideas I (Husserl 1913, §§71-72). In Formal and Transcendental Logic, Husserl expressed his satisfaction with the theory as given decades earlier (Husserl 1929, § 33).

However, while holding fast to the theory of manifolds as outlined in his published works, Husserl further refined and explained his theory in a particularly clear and explicit way in lecture courses only published decades after his death. In Einleitung in die Logik und Erkenntnistheorie (Husserl 1906-07, §§18-19) and Logik und Allgemeine Wissenschafts-theorie (Husserl 1917/18, Chapter 11), Husserl explained that he had come to discern a certain natural order in formal logic and to broaden its domain to include two layers above traditional formal logic in such a way as to account for the progress made by modern mathematics and, most particularly, the progress represented by the theory of manifolds. He considered the detection of these three layers of formal logic to be of prime importance for the understanding of logic and philosophy.

According to this new conception of formal logic, the logic of subject and predicate propositions and states of affairs, which deals with what might be stated about objects in general from a possible perspective, is found in the lowest layer. The purely logical disciplines of the two higher layers of pure logic still deal with individual things, but these are no longer empirical or material entities. They are removed from acts, subjects, or empirical persons of actual reality. For example, in the set theory of the second layer, it is not a matter of predicating something of the members, but of sets overall, having any members whatsoever (Husserl 1906-07, §18; Husserl 1917/18, Chapter 11).

In the second layer, it is no longer to be a question of objects as such about which one might predicate something, but of objective constructions of a higher kind determined in purely formal terms and dealing with objects in an indeterminate, general way. It is to be a matter of forms of judgments, and forms of their constituents, forms of deduction, forms of demonstration, sets and relationships between sets, combinations, orders, quantities, objects in general, etc. Husserl placed the fundamental concepts of mathematics in this second layer, which he conceived of as an expanded, completely developed, analytics. This is where one is to find the theory of cardinal numbers, the theory of ordinals (Ibid.).

In the second layer, one investigates what is valid for higher order objects. One proceeds in a purely formal manner since every single concept used is analytic. Here one calculates, reasons deductively, with concepts and propositions. Signs and rules of calculation suffice because each procedure is purely logical. One manipulates signs for which rules having such and such a form are valid, signs which like chess pieces acquire their meaning in the game through the rules of the game. One may proceed mechanically in this way and the result will prove accurate and justified. This is an enormous help in reasoning, for it is incomparably easier to think with signs and one is thereby freed from the equivocation and ambiguity that comes with using words. Moreover, the process itself calls for the maximum of rigor (Ibid.).

By abstracting further, one reaches Husserl's third layer, that of the theory of possible theories, the theory of manifolds. Husserl described manifolds as pure forms of possible theories which, like molds, remain totally undetermined as to their content, but to which thought must necessarily conform in order to be thought and known in a theoretical manner. In this third layer, we have a new discipline and a new method constituting a new kind of mathematics, the most universal one of all. Here formal logic deals with whole systems of propositions making up possible deductive theories. It is now a matter of theorizing about possible fields of knowledge conceived of in a general, undetermined way and purely and simply determined by the fact that they are in conformity with a theory having such a form, i.e., determined by the fact that its objects stand in certain relations that are themselves subject to certain fundamental laws of such and such determined form (Husserl 1906-07, §19; Husserl 1917/18, Chapter 11).

By using axioms of such and such a form, theories of such and such a form may be developed. These objects are exclusively determined by the form of the interconnections assigned to them. These interconnections themselves are just as little determined in terms of content as are the objects. Only their form determines them by virtue of the form of the elementary laws admitted as valid for these interconnections, laws that also determine the theory to be constructed, the form of the theories (Ibid.).

Through axiom forms, we define a multiplicity of anything whatsoever in an indeterminate, general way. After formalization words are completely empty signs that only have the purely formal sense laid down for them by the axiom forms. One then speaks of a certain something that must by definition stand in a certain relationship to something else in the defining manifold. We have concepts defined by pure, formal concepts, purely set forth, defined, as a formal possibility. Only a form is defined, but whether axioms as truths have existence in any objective real or ideal spheres corresponding to the prescribed form remains open. The form exists insofar as it is correctly defined, insofar as the axiom forms are ordered in such a way as to contain no formal contradictions, no violation of analytic principles, i.e., in the sense of formal analytical consistency. On the basis of the definition of the manifold, we can deduce conclusions, construct proofs, and it is then certain a priori that anything obtained in this way will correspond to something in our theory (Ibid.).

The general theory of the manifolds, or science of theory forms, is a field of free, creative investigation made possible once the form of the mathematical system was emancipated from its content. Once one discovers that deductions, series of deductions, continue be meaningful and to remain valid when one assigns another meaning to the symbols, one is then free to liberate the mathematical system, which can henceforth be considered as being the mathematics of a domain in general, conceived in a general and indeterminate manner. No longer restricted to operating in terms of a particular field of knowledge, we are free to reason completely on the level of pure forms. Operating within this sphere of pure forms, we can vary the systems in different ways. Nothing more need be presupposed than the fact that the objects figuring in it are such that, for them, a certain connective supplies new objects and does so in such a way that the form determined is assuredly valid for them. One finds ways of constructing an infinite number of forms of possible disciplines. And all that is of inexhaustible practical interest, Husserl maintained (Ibid.). Now let us turn to our Fregean problems.

6. Problem One: Reference to Non-existent and Impossible Objects

Gottlob Frege insisted that "we have to throw aside proper names that do not designate or name an object" (Frege 1979, 122), that "in science and wherever we are concerned about truth… we also attach a reference to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking" (Frege 1979, 118). He considered that he had shown that for certain proofs the whole cogency of the proof stands or falls on the question whether a combination of signs - e.g. Ö-1 has a meaning or not (Frege 1979, 123). In "On Formal Theories of Arithmetic", he argued that unless an equation contained only positive numbers, it no more had a meaning than the position of chess pieces expressed a truth and he condemned the theory by which one might but set down rules by which one passed from given equations to new ones in the way one moved chess pieces. A proposition might very well be non-contradictory, he stressed, without being true (Frege 1885, 112-21).

Frege could not in fact accept combinations of sign that do not designate an object because his logic was actually designed in such a way as not to be able to cope them. "Only in the case of objects can there be any question of identity (equality)", he held (Frege 1979, 182). And he put identity statements at the very heart of his project to provide foundations for the theorems of arithmetic (Frege 1884, x, §§ 55-67, 106).

It was his need for objects that had induced Frege to introduce the classes, extensions (problem two, below) that he eventually considered to be the cause of Russell's paradox (problem three, below). Consider his anguish as he wrote to Russell:

I myself was long reluctant to recognize… classes; but I saw no other possibility of placing arithmetic on a logical foundation. But the question is, How do we apprehend logical objects? And I have found no other answer to it than this, We apprehend them as extensions of concepts…. I have always been aware that there are difficulties connected with this, and your discovery of the contradiction has added to them; but what other way is there? (Frege 1979, 140-41).

In his appendix on the paradox, Frege wrote that he should gladly have dispensed with the law about extensions if he had known any substitute for it and that even then did not know how arithmetic could be scientifically established, numbers could be apprehended as logical objects without it (Frege 1980b, 214). So with Richard Heck we may well suggest that we shall "not fully understand Frege's philosophy until we understand the enormous significance the question how we apprehend logical objects… had for him…." (Heck 1995, 287)

Husserl also anguished over the logical issues surrounding combinations of signs that do not and cannot refer to objects. Such questions were a major factor in his decision to abandon work on the second volume of the Philosophy of Arithmetic, where he had planned to deal with the fractions, negative, and irrational numbers that he included under the heading of the imaginary (Hill 1991, Chapter 5 §4). Through an easy verification, he observed in 1890, one might convince oneself of the correctness of any sentence deduced by means of negative, irrational and imaginary ("impossible") numbers and after innumerable such experiences naturally come to "trust in the unrestricted applicability of these modes of procedure, expanding and refining them more and more". He complained about all the mental energy wasted that had been wasted in connection with these numbers and expressed his opinion that arithmetic would have progressed much more quickly and securely if there had there been more clarity and insight about the logic involved (Husserl 1994, 48-49).

In the foreword to the Logical Investigations, Husserl specifically alluded to having been troubled by the Mannigfaltigkeitslehre, with its expansion into special forms of numbers and extensions. The fact, he explained, that one could obviously generalize, produce variations of formal arithmetic which could lead outside the quantitative domain without essentially altering formal arithmetic's theoretical nature and calculational methods had brought him to realize that there was more to the mathematical or formal sciences, or the mathematical method of calculation than would ever be captured in purely quantitative analyses (Husserl 1900-01, 41-43; see also Husserl 1975, 35).

In a lecture that he gave before the Göttingen Mathematical Society in 1901, Husserl explained that questions regarding imaginary numbers had come up in mathematical contexts in which formalization yielded constructions which arithmetically speaking were nonsense but which could, nevertheless, be used in calculations. It was apparent, he noted, that when formal reasoning was carried out mechanically as if these symbols had meaning, if the ordinary rules were observed, and the results did not contain any imaginary components, then these symbols might be legitimately used. And this could be empirically verified. However, this fact raised significant questions and he named these three: (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 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? (Husserl 1970, 432-33; Husserl 1929, §31; Schuhmann and Schuhmann 2001).

A letter to Carl Stumpf from the early 1890s affords insight into how Husserl tackled the questions. In trying to come to a clear understanding as to how operating with contradictory concepts could lead to correct theorems, Husserl says that he had found that in the case of imaginary numbers like v2 and v -1, it was not a matter of the "possibility" or "impossibility" of concepts, for through the calculation itself and its rules, as defined for those fictive numbers, the impossible fell away, and a genuine equation remained. One could calculate again with the same signs, but referring to valid concepts, and the result was again correct. Even if one mistakenly imagined that what was contradictory existed, or held the most absurd theories about the content of the corresponding concepts of number, Husserl realized, the calculation remained correct if it followed the rules. So, he concluded that this must be a result of the signs and their rules (Husserl 1994, 15-16). In an 1891 letter to Frege, Husserl wrote that he did not have a clear idea of how Frege would justify the imaginary in arithmetic since in "On Formal Theories of Arithmetic" he had rejected the path that Husserl himself had found after much searching (Frege 1980a , 65).

Quite unlike Frege, Husserl concluded that formal constraints banning reference to non-existent and impossible objects unduly restrict us in our theoretical, deductive work. For Husserl, though no object could correspond to what was a contradiction in terms (Widersinnigkeit), a contradiction in terms nonetheless genuinely had a coherent meaning and could be determined to be true or false (Husserl 1900-01, LI IV). Unlike Frege again, Husserl found an answer that ultimately satisfied him. In Logical Investigations, he expressed his conviction that his theory of complete manifolds was the key to the only possible solution to the as yet unclarified problem as to how in the realm of numbers, impossible, non-existent, meaningless concepts might be dealt with as real ones (Husserl 1900-01, Prolegomena § 70). In Ideas, he wrote that his chief purpose in developing the theory of manifolds had been to find a theoretical solution to the problem of imaginary quantities (Husserl 1913, §72 and n.).

Understanding the nature of theory forms, Husserl explained in several texts, shows how reference to impossible objects can be justified. According to his theory of manifolds, one could operate freely within a manifold with imaginary concepts and be sure that what one deduced was correct when the axiomatic system completely and unequivocally determined the body of all the configurations possible in a domain by a purely analytical procedure. It was the completeness of the axiomatic system that gave one the right to operate in that free way. A domain was complete, according to Husserl's theory, when each grammatically constructed proposition exclusively using the language of this domain was, from the outset, determined to be true or false in virtue of the axioms, i.e., necessarily followed from the axioms (in which case it is true) or did not (in which case it is false). In that case, calculating with expressions without reference could never lead to contradictions (Husserl 1900-01, Prolegomena, § 70; Husserl 1906/07; Husserl 1917/18, §56; Husserl 1929, § 31; Husserl 1970, 441). In Ideas I, Husserl especially defined complete manifolds as having 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, he stressed, "the concepts true and formal implication of the axioms are equivalent." (Husserl 1913, §§71-72).

In Logik und Allgemeine Wissenschaftslehre, Husserl explains that one finds that there may be two valid discipline forms standing in relation to one another in such a way that the axiom system of one may be a formal limitation of that of the other. It then becomes plain that everything that can be deduced in the narrower axiom system is included in what can be deduced in the expanded system. Now, all the theorems deducible in the expanded system must exclusively contain concepts that are either valid in terms of the narrower one, and thus not imaginary, or they must contain concepts that are imaginary. Thus it is that when one compares cardinal arithmetic and ordinal arithmetic (where the minuend may be greater than the subtrahend) and their respective discipline forms, besides theorems including only non-imaginary numbers having real meaning one finds formulas and theorems that also include negative numbers, which are imaginary in terms of the more restrictive axioms of the arithmetic of cardinal numbers (Husserl 1917/18, § 56).

For example, Husserl explained, in the arithmetic of cardinal numbers, there are no negative numbers, for the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Fractions are meaningless there. So are irrational numbers, v -1, and so on. Yet in practice, all the calculations of the arithmetic of cardinal numbers can be carried out as if the rules governing the operations were unrestrictedly valid and meaningful. One can disregard the limitations imposed in a narrower domain of deduction and act as if the axiom system were a more extended one (Ibid.).

We cannot arbitrarily expand the concept of cardinal number, Husserl explained in posthumous writings on imaginary numbers. But we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, as set out in our definition, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction (Husserl 1970, 435). Fractions do not acquire any genuine meaning through our holding onto the concept of cardinal number and assuming that units are divisible, he theorized, but rather through our abandonment of the concept of cardinal number and our reliance on a new concept, that of divisible quantities. That leads to a system that partially coincides with that of cardinal numbers, but part of which is larger, --meaning that it includes additional basic elements and axioms. And so in this way, with each new quantity, one also changes arithmetics. The different arithmetics do not have parts in common. They have totally different domains, but have an analogous structure. They have forms of operation that are in part alike, but different concepts of operation (Husserl 1970, 436).

So, Husserl found, it was formal constraints requiring that one not resort to any meaningless expression, no meaningless imaginary concept that were restricting us in our theoretical, deductive work. But what is marvelous, Husserl believed, is that resorting to the infinity of pure forms and transformations of forms frees us from such conditions and at the same time explains to us why having used imaginaries, what is senseless, must lead, not to senseless, but to true results (Husserl 1917/18, § 57).

7. Problem Two: Concerning Frege's Use of Extensions

Upon learning of the paradox derivable in Basic Laws, Frege immediately spotted the source of the problem in Basic Law V, his law about extensions. He recognized its "lack of self-evidence" and confessed that he should have "gladly dispensed" with it if he had known of any substitute, but did not know how arithmetic could be scientifically established, numbers might be apprehended as logical objects unless one could pass, at least conditionally, from a concept to its extension. He said that he found it consoling that anyone else who had used extensions of concepts, classes, sets in proofs was in the same position that he was (Frege 1980b, 214).

In Basic Laws, Frege had accorded extensions "great fundamental importance", but pinpointed them as the place where any decision about defects or errors in his logic would ultimately be made (Frege 1893, vii, ix-x). He once alluded to having been "led by a certain necessity" to introduce them (Frege 1894, 197), another time to having been "constrained to overcome" his "resistance" to them (Frege 1980a, 191). He ultimately concluded that his law about extensions had indeed undermined his work, that it was to blame for the collapse of his system, that the expression 'extension of concept' easily got one into a "morass", led "into a thicket of contradictions" (Frege 1980a, 55, 130-32; Frege 1979, 269-70).

Frege had gingerly introduced extensions into Foundations of Arithmetic in order to circumvent two very specific problems: 1) the nonsensical conclusions to which his theories could lead; and 2) the sterility or unproductiveness to which they could lead (Frege 1884, §§ 66-69 and note; § 107). In Rethinking Identity and Metaphysics (Hill 1997, 61-68), Frege's reasoning is duplicated, but a more graphic example used:

Suppose that it had finally been determined to be true that the man who fired the shots from behind the grassy knoll was identical with the man who killed John Kennedy. According to Frege's reasoning, his definition of identity would then afford us a means of identifying the man who fired the shots from behind the grassy knoll again in those cases in which he is referred to as the man who killed John Kennedy. Frege recognized, however, that this means would not provide for all the cases. For example, it could not decide for us whether Lee Harvey Oswald was the man who fired the shots from behind the grassy knoll. Although it might be perfectly nonsensical to confuse Oswald with the man who fired the shots from behind grassy knoll, this would not be, according to Frege's reasoning, owing to his definition. For it says nothing as to whether the statement: 'the man who fired the shots from behind the grassy knoll is identical with Lee Harvey Oswald' should be affirmed or denied, except for the one case in which Oswald is given in the form of 'the man who killed John Kennedy'. Only if we could lay it down that if Oswald was not the man who killed John Kennedy (still parroting Frege's reasoning), could our statement be denied, while if he was that man, our original definition would decide whether it is to be affirmed or denied. But then, Frege saw, we have obviously come around in a circle. For in order to make use of this definition we should have to know already in every case whether the statement 'Lee Harvey Oswald is identical with the man who killed John Kennedy' was to be affirmed or denied.

Conscious that identities play such an important role in so many fields because we can recognize something as the same again even though it is given in a different way, Frege saw that, as it stood, his definition of number was "unproductive" because it afforded no means of recognizing that object as the same again when given differently; it presupposed that an object could only be given in one single way. All identities, he realized, would then amount simply to the principle "so obvious and sterile as not to be worth stating" "that whatever is given in the same way is to be reckoned as the same and one could not draw from it any conclusion which was not the same as one of the premises" (Frege 1884, §§ 66-67; § 107).

Returning to the Kennedy example, suppose that we finally have incontrovertible evidence that a certain Mr. Knoll shots at Kennedy from behind the grassy knoll and those shots killed him. 'Mr. Knoll is identical with Kennedy's assassin' is then a true statement and we should then be able substitute Mr. Knoll's name every time we find a reference to Kennedy's assassin. However, our identity statement is so blind as concerns other alternatives and, in this case Kennedy's assassin has so often been identified with Oswald, that substitution in most contexts would yield sheer nonsense.

In his irresponsible review of the Philosophy of Arithmetic, Frege chastised Husserl for not having used the term 'extension of a concept'. Any view, Frege complained, according to which a statement of number is not a statement about a concept or about the extension of a concept is naive for "when one first reflects on number, one is led by a certain necessity to such a conception" (Frege 1894, 197, 201-02). Nonetheless, Husserl never seems to have been tempted by extensions. In Formal and Transcendental Logic, we still find him calling extensional logic naive, risky, doubtful and its fundamental unclarity the source of many a contradiction requiring every kind of artfulness to make it safe for use in reasoning (Husserl 1929, §§ 23b, 26c).

In Philosophy of Arithmetic, Husserl wrote that he did not see how Frege's method might signify an enrichment logic. What Frege's method actually enabled one to define, Husserl found, was the extension of the concepts. All Frege's "definitions become correct propositions if one substitutes extensions for the defining concepts, but of course become completely obvious and worthless propositions", Husserl deemed. For him, Frege's theory showed that:

the "direction of the straight line a is the extension of the concept 'parallel to straight line a'". By extension of a concept, though, one means the aggregate of the objects falling under it. The direction of the straight lines a would thus be the aggregate of straight lines parallel to a. Likewise, it showed that "the shape of the triangle d is the extension of the concept 'similar to triangle' d", meaning the aggregate of all the triangles similar to d. And "the number belonging to concept F" is, therefore, it too defined in this manner as the concept "numerically equal to concept F". In other words: the concept of this number is the whole of the concepts numerically equal to F, therefore a whole of infinitely many "equivalent" sets. (Husserl 1891, 132)

According to Frege's definition, Husserl observes, 'number of Jupiter's moons' would accordingly mean "having the same number as the concept Jupiter's moons", or more clearly expressed "having the same number as the aggregate of Jupiter's moons". Thus one obtains concepts having the same extensions, but not the same intension. The latter concept is identical with the concept "any set whatsoever from the equivalence class determined by the aggregate of Jupiter's moons." All these sets also fall under the number four. That different concepts are there requires no proof, Husserl considers.

Foreshadowing ideas about replacing a calculus of classes with a calculus of concepts that Husserl ultimately espoused, he added that Frege himself seems to have sensed the doubtfulness of his definition since in a note he says of it that he believes that instead of "extension of concept one could simply say 'concept'" (Husserl 1891, 134-35, n. 1). By 1890, Husserl may have actually been permanently inoculated against recourse to extensions. Several articles published during the 1890s find Husserl laying bare "the follies of extensional logic" (Husserl 1994,199), which he would replace by a calculus of concepts (Husserl 1994, 92-114, 115-20, 121-30, 135-38, 443-51).

Husserl wrote quite a bit about how extensions, objects, totalities, aggregates, sets, and manifolds stand in relationship one to the other in his unpublished notes on set theory. There his objections to extensions and his espousal of manifolds turn on his convictions that objects stand in relations with respect to certain properties and that among these relations are those belonging to the essence (Ms A 1 35, 10a). Husserl's manifolds are not aggregates of elements without relations. It is precisely the relations that are essential and serve to distinguish a manifold it from a mere aggregate. As explained above, Husserl saw manifolds is aggregates of elements that are not just combined into a whole, but are continuously interdependent and ordered so that each member possesses an unambiguous position in relation to any other one (Husserl 1983, 93, 95-96, 410). The properties and relations set out in the axioms of a complete manifold in Husserl's sense determine objects unequivocally, bring information and logically eliminate nonsensical conclusions, something that Frege hoped to do within his one theory by resorting to extensions.

9. Problem Three: The Contradictions of Set Theory

Of course, the really big thorn in Frege's side turned out to be the contradiction derivable in the system of Basic Laws. Husserl characterized what he referred to as "Russell's paradox" as follows:

All sets fall into such that contain themselves as an element, and such that do not do this. What is the set of sets that does not contain itself as an element to be viewed as being? If it does not contain itself as an element, then it is not the set of all sets that do not contain themselves. If it, however, does contain itself, then it is among the sets of all sets that do not contain themselves as element, one that contains itself as an element. (Ms A 1 35, 26a)

Of what he referred to as "Zermelo's paradox", Husserl wrote:

Zermelo argues: A set M that contains each of its partial sets as elements is an inconsistent set. 1) We consider those partial sets that do not contain themselves as elements. 2) In their entirety these form a set Mo that is contained in M. 3) Mo is thus an element of M. 4) Mo is not an element of Mo. Proof: were Mo an element of Mo, then it would contain a partial set of M (namely Mo) that contains itself as element. However, Mo is to contain ex definitione partial sets of M that do not contain themselves as elements. 5) Thus Mo, since is not an element of Mo, is a partial set of M, which does not contain itself not as element. But all such sets are ex definitione contained in the concept of Mo, thus in opposition to 4. But Mo is an element of Mo. We come to a direct contradiction. If it essentially belongs to the concept of set that (without contradiction) no set can contain itself as an element, then Mo and M are identically the same set, and we show that the whole reasoning was untenable. (Ms A 1 35, 24a)

The telling phrase here is: "If it essentially belongs to the concept of set that (without contradiction)…." Given Husserl's conviction that logical, mathematical laws are laws of essence, it is not surprising to find him arguing over and over that the set-theoretical paradoxes must involve some violation of the essence of set.

His notes on set theory record his reflections about just what the essence, the concept, of set entails. It is part of the idea (Idee) of the set to be a unit, a whole comprising certain members as parts, but doing so in such a way that, vis-à-vis its members, it is something new which is first formed by them. All mathematico-logical operations performable with sets, he wrote, turn on the idea that sets can be looked upon as kinds of wholes, as new units, formations that are something new vis-à-vis their original members, so that out of these formations new units can then again be formed. The unity of a system is something new vis-à-vis the elements systematized. It would be an contradiction in terms for the system's unity itself to be able to be one among the elements of the same system, upon those the system bases itself. But system units can themselves again be systematized and then ground higher forms of system. They then, however, bring elements into a new system whole (Ms A 1 35, 20b). Wherever mathematicians speak of sets, Husserl maintained, if the concept is to be a mathematical one, they must have a set essence in view, and whatever sets may have as an essence, it is expressed with a relation that belongs to the essence, i.e., the relation between sets themselves and elements of a set (Ms A 1 35, 12b).

If the formal logical construction "set of all sets which do not contain themselves as parts" is considered, it may not be presupposed that they already exist, i. e. that the extension of the conception produces a totality without further ado (Ms A 1 35, 43a). From the fact that one can speak of all sets, it does not follow that the totality of sets can in return be looked upon as a set (Ms A 1 35, 20b). Mere talk like "a set which contains each of its partial sets as an element" guarantees nothing. One can initially, upon occasion, avoid talking about sets in general. But then one is bound to logic, which requires that sets, genuine manifolds must come under the formal rules for wholes and that consistency be there (Ms A 1 35, 21).

"It belongs essentially to the concept of set that (without contradiction) no set can contain itself as an element", Husserl reiterated constantly (Ms A 1 35, 24a). An essence relation, , i.e., the relation between sets themselves and elements of a set, makes it impossible for the members of the relation to be identical. Hence a set that contains itself as a element would be a contradiction in terms (Widersinnigkeit) (Ms A 1 35, 12b). A whole cannot be its own part. Just as it is contradictory for a whole to be its own part at the same time, so it is contradictory for a set for it to be its own member (Ms A 1 35, 20b). To the objection that there is no set that contains itself as an element, he maintained that one need merely respond that that is a contradiction in terms (Widersinnigkeit) (Ms A 1 35, 17a).

Husserl thus repeatedly relegated the set theoretical paradoxes to the category of contradictions in terms (Widersinnigkeiten), as defined in the Fourth Logical Investigation where it is the job of logical laws to guard against formal or analytical contradictions in terms, formal absurdity, by dictating what it is that objects require to be consistent in purely formal terms (Husserl 1900-01, LI IV Introduction; § 12). If one is clear and distinct with respect to meaning, Husserl suggested in his unpublished notes, one readily sees the contradiction in terms (Widersinnigkeit) involved in the set-theoretical paradoxes. So, the solution to the paradoxes would then lie in demonstrating the shift of meaning that makes it that one is not immediately aware of the contradiction in terms and that once one perceives it, one cannot indicate wherein it lies (Ms A 1 35, 12a).

All sets, one finds Husserl reasoning in his unpublished notes, fall into one of two classes. Either they fall into class A of sets that contain themselves as an element, or they fall into class B of sets that do not. That being so, he asked to which of these two classes does the set PM of all sets that do not contain themselves as an element belong? By the law of the excluded middle, it would have to be one of the two. However, he observed, one can show that neither or both of the possibilities must be valid. For were PM to belong to class A, that would mean that it contained itself and that would contradictory by definition. Were it fall into class B, PM would then not be the set of all sets of all Bs. In actual fact, it proceeds from the paradox, if no conceptual shift is demonstrable, that a set of kind A or a set of kind B must be a contradiction in terms. Then the classification is a contradiction in terms as well. He then looks at the alternative that both may not be the case and asks whether PM is not a contradiction in terms (Ms A 1 35, 17a).

Nonetheless, although Husserl's notes on set theory record many a reflection on what the concept of set entails, he also expresses his conviction there that we do not yet by any means have the real and genuine concept of set that the mathesis and logic need. The paradoxes, he declared, only demonstrate that a general logic of sets in general, of totalities is still lacking. He stressed that in his logic courses he had constantly from the beginning said that totality and set should not be identified and that this identification must be partly responsible for the paradoxes of set theory (Ms A 1 35, 43a, 69a).

Now Husserl's theory of Mannigfaltigkeiten was precisely a theory about how to engage in pure a priori analyses of essence. His unpublished notes on set theory show him suggesting reforming the mathematical theory of manifolds by consciously transforming it into a transcendental theory of manifolds that consciously captures the formal essence of a genuine, constructible totality, consciously analyzes what belongs to the essence of a concept defining a totality, what belongs to the essence of an axiom and axiom system that as such establishes the univocity and construction and only establishes the meaning of what is constructed, which then formally weighs the conditions of the possibility of such a totality and by this means derives the system of possible totality forms, or manifold forms, so that it inquires into the possible forms of construction principles and into the formal system in which they themselves can be constructed as forms (Ms A 1 35, 38). Each field of an exact science, he maintained, is a constructible totality. For that field "axioms" are valid in which construction principles for the totality must lie and they make a systematic thinking and relationships of genuine arguments and genuine inferences, apodictic conclusions possible (Ms A 1 35, 45a). The validity and non-validity of all derivable concepts are to be decided in accordance with the axioms and they are to delimit the talk of a manifold of "existing" formations (Ms A 1 35, 13a).

Of course, the kinds of contradictions in terms flagged above would no longer have any place in such a theory, for which consistency and completeness were paramount. As someone who frequented Hilbert's school in Göttingen, and who upon several occasions pointed to the kinship existing between his own manifolds and Hilbert's axiomatic systems (Husserl 1913, § 72; Husserl 1929, § 31), Husserl seems to have concluded that once one indeed had the real and genuine idea of set needed, once could proceed with the axiomatization of set theory in a way compatible with his own theories about sets and manifolds. Needless to say, he did not believe that Frege's system of axioms, with its law about extensions, could embody the real and genuine idea of set needed.

10. Conclusion

Much more can be said about the significance of Husserl's theories about sets and manifolds than can be said here. Here I have set out the main steps in the development of those ideas and I have tried to put some flesh on Husserl's little explored theory of manifolds by showing how it developed in connection with some specific problems that Husserl perceived in set theories espoused by his contemporaries, --including his own. My discussion focussed on three Fregean problems that Husserl explicitly addressed. I defined certain issues involved and provided information about Husserl's theory of manifolds that provides clues as to how it might confront them. May what I have written here help bring Husserl's ideas about sets and manifolds out of the realm of abstract theorizing and prompt further exploration of this philosophical territory, which is as uncharted as it is rich in philosophical implications needing to be drawn and to be made known.


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