This is a preprint version of the paper which appeared in definitive form in Husserl or Frege, Meaning, Objectivity and Mathematics, by Claire Ortiz Hill and Guillermo Rosado Haddock, Chicago: Open Court, 2000 and Idealization IV. Historical Studies on Abstraction and Idealization Poznan studies in the philosophy of the sciences and the humanities vol. 82, F. Coniglione, R. Poli, R. Rollinger (eds.), Rodopi, Amsterdam, 2004, pp. 217-43. The published versions should be consulted for all citations. 

 

ABSTRACTION AND IDEALIZATION

IN EDMUND HUSSERL AND GEORG CANTOR PRIOR TO 1895

by CLAIRE ORTIZ HILL

 

After two years of training in philosophy under Franz Brentano, Edmund Husserl arrived in Halle, Germany in 1886 with a plan to help provide radical foundations for mathematics by engaging in psychological analyses of the concept of number. To this end, he espoused a theory of abstraction of the concept of number akin to the one his colleague Georg Cantor was propounding in conjunction with the Platonistic theories about sets he was hard at work developing and defending.

However, Husserl rather quickly grew profoundly dissatisfied with the psychological foundations he had begun laying for mathematics in his On the Concept of Number (Husserl 1887) and then in the Philosophy of Arithmetic (Husserl 1891). He even confessed to having experienced doubts about his approach right from the very beginning (Husserl 1913, 34-35). The first clues to the answers to the many questions his efforts to provide more secure foundations arithmetic raised would ultimately be found in Hermann Lotze's theory of Platonic ideas (Husserl 1913, 36-42, 44-47). And during years of hard, solitary work in Halle Husserl developed the original interpretation of Platonic idealism which went into the making of the phenomenological method (Husserl 1900-01; Husserl 1913).

Very little is known of Husserl's encounter with the Georg Cantor's theories about Platonic idealism and the abstraction of number concepts. Much, however, can be gleaned about this from a close study and comparison of Cantor's and Husserl's writings during those crucial years in Husserl's development. So in the following pages I try to shed light on that dark period in Husserl's development by studying the evolution his ideas underwent as this relates to Cantor's philosophizing about abstraction, Platonic idealism and the concept of number. I focus on the important changes which took place in Husserl's ideas during his first ten years in Halle.

1. Creating Numbers in the Mannigfaltigkeitslehre

During Husserl's fifteen year sojourn in Halle, Georg Cantor was hard at work laying the foundations of set theory and reconnoitering, conquering, colonizing and defending the new world of transfinite numbers. Any further progress of his work on set theory, he had explained in the beginning of his Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Cantor 1883), was absolutely dependent upon the expansion of the concept of real whole numbers beyond the present boundaries and in a direction which, as far as he knew, no one had yet searched. He had, he claimed, burst the confines of the conceptual formation of real whole numbers and broken through into a new realm of transfinite numbers. As strange and daring as his ideas might now seem, he was convinced that they would one day be deemed completely simple, appropriate and natural (p. 165). Much of the work he would do in the coming years would be aimed at showing they were.

Upon discovering the transfinite numbers, Cantor tells readers of the Mannigfaltigkeits-lehre, he had not been clearly conscious of the fact that these new numbers possessed the same concrete reality the whole numbers did. He was, however, now persuaded that they did (p. 166) and intent upon proving that "after the finite there is a transfinite... which by its very nature is not finite, but infinite, which, however, can be determined by numbers which are definite, well-defined, and distinct from one another just as the finite can" (p. 176).

Acting on a conviction, spelled out in a 1884 letter to Gösta Mittag-Leffler, that the only correct way to proceed was "to go from what is most simple to that which is composite, to go from what already exists and is well-founded to what is more general and new by continually proceeding by way of transparent considerations, step by step without making any leaps" (Cantor 1991, 208), Cantor began devising a strategy as to how to provide his "strange" new numbers with secure foundations by demonstrating precisely how the transfinite number system might be built from the bottom up.

Through the combined action of two principles, he argued in the Mannigfaltigkeitslehre, one might break through any barrier in the conceptual formation of the real, whole numbers (pp. 166-67) and with the greatest confidence and self-evidence arrive at ever new number classes and numbers having the same concrete definiteness and reality as objects as the previous ones (p. 199). In §§ 1, 11 and 12 of the work he showed how his principles might lead to the definitions of the new numbers and produce number classes. By the first principle, the first series of positive real whole numbers 1,2,3,..., v would have their origin and basis in the repeated positing and adding of underlying units considered to be identical, the number v being the expression both for a specific finite number of units posited in this way and for the joining of the posited units into a whole. Numbers thus manufactured would be members of the first number class which was infinite and for which there was no greatest number (p. 195). By the second principle, whenever there was any definite succession of defined whole numbers for which no greatest one existed, a new number could by created considered as the boundary of those numbers, i.e. defined as the next greater number to them all (pp. 195-96). By a third principle, the second number class defined would not only receive a higher power than the first, but precisely the next higher one, hence the second power (p. 167; 197). By "power" Cantor meant cardinal number.

In a note to the Mannigfaltigkeitslehre Cantor also provided a rather inchoate account of his idea of the procedure for the correct formation of concepts:

One lays down a propertyless thing, which is at first nothing other than a name or a sign A, and systematically gives it different, even infinitely many distinct predicates whose meaning is generally known through already existing ideas and which may not contradict each other; through this, A's relations to already existent concepts, and in particular to related concepts, will be determined. When one has completely finished with this, then all the conditions for awakening the concept A which was slumbering in us are present and it comes into existence... (p. 207 notes 7, 8).

2. Cantor's Theory of Abstraction

One of the many questions Cantor's Mannigfaltigkeitslehre account of concept formation raises is that of how exactly one manages to lay hold of the propertyless objects by which number concepts are to be manufactured. The answer, though, was forthcoming. For since 1883, Cantor had begun making his thereafter oft repeated claims that his transfinite numbers could be "produced through abstraction from reality with the same necessity as the ordinary finite whole numbers by which alone all other mathematical conceptual formations thus far been produced" (Cantor 1991, 136; Kreiser 1979).

Cantor's earliest attempt to publish his theory of abstraction, however, met with failure (Cantor 1884; Cantor 1991, 226-30, 240-44). In the posthumously published "Principien" of 1884 he had written:

The power of a set M is hereupon defined as the presentation of what is common to all of the sets M of equivalent sets and only those and hence also of the set M itself; it is the representatio generalis... for all sets of the same class as M. It therefore seems to me to be the most primitive, pyschologically, as well as methodologically simplest root concept, arisen through abstraction, from all particular characteristics which a set of a specific class may display, both with respect to the nature of its elements, as well as with regard to the relations and order in which the elements are to each other or can stand to things lying outside the set. The concept of power originates in reflecting only upon what is in common to all of one and the same class of member sets (Cantor 1884, 86).

Accounts of this very process appear in Cantor's correspondence of the time (ex. Cantor 1991, 178-80; Eccarius 1985). And when the theory was first published in the "Mitteilungen zur Lehre vom Transfiniten" (Cantor 1887/8), Cantor would stress that he had advocated and repeatedly taught it in his courses as much as four years earlier. Integrated directly into the "Mitteilungen" are parts of letters he had written on the subject during the years prior to the publication of the work (pp. 378-79, 387 n., 411 n.).

When the abstraction theory did find its way into print, it formed an integral part of Cantor's endeavor to provide solid foundations for transfinite arithmetic in the "Mitteilungen" (Cantor 1887/8), where Cantor particularly sought to show how his theorems about transfinite numbers were firmly secured "through the logical power of proofs" which, proceeding from his definitions, which were "neither arbitrary nor artificial, but originate naturally out of abstraction, have, with the help of syllogisms, attained their goal" (p. 418). Much of the "Mitteilungen" is, in fact, devoted to explaining exactly how numbers are to be procured from reality by abstraction and, in particular, how the actual infinite number concept is to be formed through appropriate natural abstractions in the way the finite number concepts are won through abstraction from finite sets (p. 411; Cantor 1991, 329, 330).

For Cantor, cardinal numbers were, as he explained to Giulio Vivanti in 1888, the general concepts assigned to sets which one may obtain by abstracting both from the properties of the elements and from the order in which they are given (Cantor 1991, 302). In the "Mitteilungen" he repeatedly gave essentially the same recipe for extracting cardinal numbers from reality through abstraction:

In abstracting from a given set M, composed of determinate, completely distinct things or abstract concepts called the elements of the set and thought of as a thing for itself, both the properties of the elements and the order in which they are given, is produced in us a determinate general concept... which I call the power of M or the cardinal number corresponding to the set M (p. 411; see also pp. 379, 387).

There he also provided this concrete illustration of the procedure he was prescribing:

For the formation of the general concept "five" one needs only a set (for example all the fingers of my right hand) which corresponds to this cardinal number; the act of abstraction with respect to both the properties and the order in which I encounter these wholly distinct things, produces or rather awakens the concept "five" in my mind (p. 418 n. 1).

Cantor believed his theory of abstraction to be the distinctive feature of his number theory and that it represented an entirely different method for providing the foundations of the finite numbers than was to be had in the theories of his contemporaries. As he wrote to Giuseppe Peano:

my view of "numbers" is fundamentally toto coelo different from those we find in Grassmann, Dedekind, Helmholtz, Weierstrass, Kronecker, etc.; and from this results, as you see from my work, a completely different method for the grounding of "finite number theory". With those conceptions one would have never have come upon the transfinite numbers whose grounding is only possible in the way which I have carved out (Cantor 1991, 365; 363).

With his theory as to how number concepts might be abstracted from reality, Cantor explained in the "Mitteilungen", he was laying bare the roots from which the organism of transfinite numbers develop with logical necessity (Cantor 1887/8, 380). He said that it was the thought process involved in forming the actual infinite number concept through abstraction like that by which finite number concepts were won from finite sets that had led him to the theory of transfinite numbers he had begun to outline in the Mannigfaltigkeitslehre (Cantor 1887/8, 411).

The definition by abstraction would eventually make its way into the Beiträge where Cantor maintained that the principles he had "laid down, and on which later the theory of the actually infinite or transfinite cardinal numbers will be built, afford also the most natural, shortest, and most rigorous foundation of the theory of finite numbers" (Cantor 1895/7, § 5).

Since Cantor was propounding this theory of abstraction at the very time Husserl was writing On the Concept of Number (1887) and the Philosophy of Arithmetic (1891) where a quite similar theory was advocated, it is important to note here that in the latter work, Husserl approvingly noted that Cantor's definition of number in the "Mitteilungen" is profoundly different from that of the Mannigfaltigkeitslehre (Husserl 1891, 126 n.) . Husserl specifically points to two passages of the "Mitteilungen". In the first one, Cantor had written:

By the power or cardinal number of a set (Menge) M (which is made up of distinct, conceptually separate elements m, m',... and is to this extent determined and limited), I understand the general concept or species concept (Gattungsbegriff) (universal) which one obtains by abstracting from the properties of the elements of the set, as well as from all the relations which the elements may have, whether themselves or to other things, but especially from the order reigning among the elements and only reflect upon what is common to all sets equivalent to M (Cantor 1887/8, 387; also Cantor 1991, 178).

Calling Cantor a mathematician of genius and referring to the above cited passage concerning the formation of the general concept "five", Husserl further commends Cantor for having written with a great deal of precision in the "Mitteilungen" that for "the formation of the general concept "five" one needs only a set... which corresponds to this cardinal number" (Husserl 1891, 126 n.).

3. A Detour by Way of Gottlob Frege and Louis Couturat

Two of Cantor's and Husserl's contemporaries, Gottlob Frege and Louis Couturat, offered contrasting interpretations of the abstraction theory the Halle men favored, the judgements they passed affording valuable insight in the nature of the philosophical issues involved in Cantor's and Husserl's attempts to win numbers from reality through abstraction.

In his cranky reviews of Cantor's "Mitteilungen" (Frege 1892; Frege 1979) and Husserl's Philosophy of Arithmetic (Frege 1894), Frege charged both men with attempting to achieve magical effects by by using abstraction to destroy the properties things have (Frege 1979, 69-70; Frege 1894, 205). Abstraction, Frege said, would endow mathematicians with the miraculous, supernatural ability (Frege 1979, 69, 71) to change things (Frege 1979, 70; Frege 1894, 197-98, 204-05) in "the wash-tub of the mind" (Frege 1894, 205; Chapter 6 of this book).

Frege himself held that the "properties which serve to distinguish things from one another are, when we are considering their Number, immaterial and beside the point" and that "we want to keep them out of it" (Frege 1884, § 34); according to him one was to "follow pure logic" by "disregarding the particular characteristics of objects" (Frege 1879, 5). He furthermore believed that the propositions of pure logic could not "come to consciousness in a human mind without any activity of the senses" since "without sensory experience no mental development is possible in the beings known to us" (Frege 1879, 5). But it was the "psychological and hence empirical turn" (Frege 1892, 180, 181) he believed Cantor and Husserl had given the matter that particularly irked him (Frege 1894, 197, 208-09).

Couturat interpreted Cantor's and Husserl's efforts otherwise. In his criticism of empirico-psychologistic theories of number in De l'infini mathématique (1896), he argued that, despite claims by Leopold Kronecker and Hermann von Helmholtz (as well as Richard Dedekind and Carl Friedrich Gauss) that their theories of number were "pure of any worldly blemish and any commerce with the world of the senses" (p. 319), their theories of number were in reality empiricist (pp. 318-31) because they just described the psychological process of counting without inquiring into the ideal conditions which make it possible and intelligible (p. 331). According to their theories, Couturat explains, the concept of whole cardinal number would originate in experience, in the counting of a concrete set of external objects given in perception (p. 319; p. 329). The two words "psychologistic" and "positivistic", he says, sum up and display the inadequacy and vice of such an empirical approach (pp. 330-31). And, interestingly, he praises Husserl (p. 331 n.) for having come to that very conclusion in the Philosophy of Arithmetic.

The elements of a collection cannot, Couturat objects, be counted as concrete objects, but only, as Cantor does, as a set of abstract, propertyless units obtained when by "abstracting from the particular nature of the objects given and from their distinctive properties, one considers each one of them as one, meaning one reduces it to a unit, and embraces all those abstract units in a single mental act.…" (pp. 325-26)

So for Couturat, Cantor's theory of number is only "empiricist in appearance", but is "rationalist in reality" (p. 332; p. 335) because it is based on the "genuinely a priori" (p. 332) "metaphysical idea of unit" (p. 341), which could not be the residue of any abstraction performed on sensory data. For a unit is neither a perception nor anything given in perception (p. 340), but is "a pure rational form which the mind imposes a priori on all its objects just by the fact that it thinks them" (p. 341).

Such an interpretation lends insight into the nature of Cantor's theories and into the development of Husserl's ideas in the same area. It is in fact easy to find passages in Cantor's work where any appearance of empiricism or psychologism is belied by an underlying rationalism. For, however empirical or psychological Cantor's mysterious references to inner intuition (ex. Cantor 1883, 168, 170, 201), or to experiences helping produce concepts in his mind (Cantor 1887/8, 418 n. 1) may appear, he believed abstraction would liberate mathematicians to engage in strictly arithmetical forms of concept formation by freeing them from psychologism, empiricism, Kantianism and insidious appeals to intuitions of space and time (ex. Cantor 1883, 191-92; Cantor 1885; Cantor 1887/8, 381 n. 1; Eccarius 1985, 19-20).

In speaking out in the Mannigfaltigkeitslehre against the new empiricism, sensualism, skepticism and Kantianism which, he argued, mistakenly located the sources of knowledge and certainty in the senses or in the "supposedly pure forms of intuition of the world of presentation", Cantor had affirmed that sure knowledge could "only be obtained through concepts and ideas which, at most stimulated by external experience, are on the whole formed through inner induction and deduction as something which in a way already lay within us and was only awakened and brought to consciousness" (p. 207, note 6).

4. Cantor and Platonic Ideas

Cantor surely made no secret of his intention to supply his numbers with adequate philosophical and metaphysical foundations. The Mannigfaltigskeitslehre (1883) came replete with epistemological and metaphysical reflections aimed at explaining and justifying his novel ideas to a readership chary of such talk (ex. Cantor 1991, 100, 113, 118, 178, 199, 227). In the beginning of the Principien, expressing the high regard he had for metaphysics and his belief in a close alliance between metaphysics and mathematics, he thanked Jules Tannery for having paid him the honor of according philosophical, and even metaphysical, worth to his writings (Cantor 1884, 83-84). In 1885, Mittag-Leffler even felt the need to warn Cantor that his new terminology and philosophical way of expressing himself might be so frightening to mathematicians as to seriously damage his reputation among them (Cantor 1991, 241). The "Mitteilungen" (1887/8) that Cantor published in a philosophical journal largely represented an attempt on his part to provide philosophical justification for his new numbers, a concern which fills the pages of his letter books as well (Cantor 1991).

Cantor was also most explicit about the precise nature of the metaphysical underpinnings he was proposing for his numbers. He believed that the transfinite "presented a rich, ever growing field of ideal research" (Cantor 1887/8, 406) and saw abstraction as showing the way to that new, abstract realm of ideal mathematical objects which could not be directly perceived or intuited. His talk of awakening and bringing to consciousness the knowledge, concepts and numbers slumbering in us (Cantor 1883, 207 n. 6, 7, 8; Cantor 1887/8, 418 n. 1) is an unmistakeable allusion to Plato's theory of recollection and Socratic theories of concept formation (ex. the Meno 81C-86C; Phaedo 72E, 75E-76A). He considered his transfinite numbers to be but a special form of Plato's arithmoi noetoi or eidetikoi, which he thought probably even fully coincided with the whole real numbers (Cantor 1884, 84; Cantor 1887/8, 420). To Giuseppe Peano he wrote: "I conceive of numbers as 'forms' or 'species' (general concepts) of sets. In essentials this is the conception of the ancient geometry of Plato, Aristotle, Euclid etc." (Cantor 1991, 365). By manifold or a set, he wrote in the Mannigfaltigkeitslehre, he was defining something related to the Platonic eidos or idea, as also to what Plato called a mikton in the Philebus (Cantor 1883, 204 n. 1). "My idealism", he wrote to Paul Tannery, "is related to the Aristotelian-Platonic kind, which as you know is at the same time a form of realism. I am just as much a realist as an idealist" (Cantor 1991, 323). In the Mannigfaltigkeitslehre he had emphasized that the "certainly realist, at the same time, however, no less than idealist foundations" of his reflections were essentially in agreement with the basic principles of Platonism according to which only conceptual knowledge in Plato's sense afforded true knowledge, but that the nearer our presentations came to the truth, the nearer their objects must come to being real and vice versa (Cantor 1883, 181, 206 note 6).

However, despite his explicit appeals to Platonism and his multiple references to philosophical writings, Cantor did not explicitly refer readers to any of Plato's writings other than the Philebus (Cantor 1883, 204 note 1). He did, though, refer them (ex. Cantor 1883, 206 note 6; 205 note 2) to Eduard Zeller's books on Greek philosophy (Zeller 1839; Zeller 1875; Zeller 1879), which Cantor's "frightening" new terminology and philosophical way of expressing himself so echoes as to afford insight into the precise nature of Cantor's understanding of the Platonic Ideas he was espousing. So examining the sixty pages of Zeller's synthesis of Plato's doctrines (Zeller 1875, 541-602) that Cantor directly cited in the Mannigfaltigkeitslehre (Cantor 1883, 206 n. 6) --much of which is a standard interpretation of Plato's writings-- adds to what we can harvest about Cantor's Platonism from the spare remarks he scattered throughout his writings.

For one thing, Zeller's account of Platonic Ideas sheds some light on how Cantor might have come to think it reasonable to marry a theory of abstraction with a theory of sets as Platonic eidos or idea and numbers as Platonic eidetikoi. For the formation of concepts through abstraction is a process generally associated with Aristotelian and empiricist philosophy and is generally considered to be incompatible with the basic principles underlying a strictly Platonic theory of forms or Ideas known through recollection (Weinberg 1968, 1). Aristotle himself made no secret of his opposition to Plato's Ideas and eidetikoi (Metaphysics Books M-N). Given these facts any attempt to marry abstraction and Platonic Ideas might be viewed as crude and uncouth.

Yet we find Cantor maintaining that his technique for abstracting numbers from reality provides the only possible foundations (Cantor 1991, 365, 363; Cantor 1887/8, 380, 411) for his Platonic conception of numbers. Knowledge, he maintained, could "only be obtained through concepts and ideas which, at most stimulated by external experience, are on the whole formed through inner induction and deduction as something which in a way already lay within us and was only awakened and brought to consciousness" (Cantor 1883, 207 note 6). By engaging in the process of concept formation prescribed in the Mannigfaltigkeitslehre, Cantor maintained one fulfilled the conditions for "awakening" a concept which "slumbering in us" "comes into existence" (Cantor 1883, 207 notes 7, 8). And it was his theory of abstraction that was going to yield those concepts and ideas.

Zeller, however, actually attributed a theory of abstraction to Plato, giving some clue as to how Cantor might have justified his own interpretation. "The Ideas", Zeller wrote in pages Cantor cited (Cantor 1883, 206 note 6), "arisen out of Socratic concepts, are in fact, like these, abstracted from experience, however little Plato is willing to have this word; they thus first present a particular, and only step by step move up from this particular to the general, from lower to higher concepts" (Zeller 1875, 584). Plato, Zeller contended, was "not aiming for a pure a priori construction, but only for a complete logical ordering of the Ideas, which he himself found through induction, or if we prefer: through an increasing recollection of what is sensory" (Zeller 1875, 584).

In so interpreting Plato, Zeller was certainly thinking of passages from the Dialogues in which there is talk of passing from a plurality of perceptions to a unity gathered together by reasoning (ex. Phaedrus 249B-C), or in which Plato wrote of using of our senses in connection with objects to recover or recollect previously acquired knowledge (ex. Phaedo 75E). The soul, Socrates would demonstrate in the Meno, has learned everything and nothing prevents someone from discovering everything because searching and learning are recollection (Meno 81C-D,) so that those who do not know have within themselves true opinions about the things they do not know, and these opinions can be stirred up like a dream in such a way that in the end their knowledge about these things would be as accurate as anyone's (Meno 85C).

Cantor directly appealed to Zeller's interpretation of Plato's theories in a note (Cantor 1883, 206 note 6) to the important section of the Mannigfaltigkeitslehre in which the Halle mathematician explains and justifies his conviction that "mathematics is entirely free in her development and only bound to the obvious consideration that her concepts both be not self-contradictory and stand in ordered relationships fixed through definitions to previously formed, already existing and proven concepts" (Cantor 1883, 182).

This freedom, Cantor maintained, was a consequence (Cantor 1883, 182) of the connection between the two kinds of reality or existence which both finite and infinite whole numbers enjoy and which, strictly speaking, they share with any concepts or Ideas whatsoever. First, Cantor maintained, these numbers have intrasubjective or immanent reality "inasmuch as on the basis of definitions they occupy a fully determinate place in our understanding, are as distinct as possible from all other components of our thought, stand in determinate relations to them and consequently modify the substance of our mind in determinate ways" (Cantor 1883, 181). A second kind of reality may, Cantor believed, also be ascribed to these numbers. They may have transsubjective or transient reality inasmuch as "they must be regarded as an expression or image of processes of the external world lying outside of the intellect, as further the different number classes... are representatives of powers which are actually present in corporeal and intellectual nature" (Cantor 1883, 181).

The "thoroughly realistic, but no less idealistic, foundation" of these reflections, Cantor explains, leaves no doubt in his mind "that these two kinds of reality come together constantly in the sense that a concept existing in the first sense always also possesses transient reality in certain, even infinitely many respects..." (Cantor 1883, 181). "Only conceptual knowledge", Cantor now cites Zeller, "is said (according to Plato) to afford true knowledge. The nearer, however, our presentations come to the truth... the nearer their objects must come to being real and vice versa. What is knowable, is; what is not knowable, is not, and to the same extent something is, it is also knowable (Cantor 1883, 206-07 note 6; Zeller 1875, 541-42).

As a consequence, Cantor concludes, mathematics has "purely and simply to take into consideration the immanent reality of her concepts and hence no obligation whatsoever to investigate their transient reality". "In particular", he continues, "with the introduction of new numbers she is only dutybound to give definitions of them that bestow upon them such determinacy and, if need be, such a relationship to the older numbers, that in given cases they may be distinctly differentiated from one another. As soon as a number meets these conditions, it can and must be considered as existent and real in mathematics". This is the reason, Cantor maintains, "why one is to regard the rational, irrational, and complex numbers as altogether just as existent as the finite positive whole numbers". The actual basis of the connection between these two kinds of reality, he tells us, lies "in the unity of the universe, to which we ourselves belong" (Cantor 1883, 182).

Unity was in fact a major theme of Cantor's philosophy of arithmetic. Though he maintained that to be considered existent and real in mathematics numbers must be must distinctly differentiated from one another and as distinct as possible from all other components of our thought, he stressed over and over that these independent numbers in and for themselves organically coalesce into a unified whole in special ways (Cantor 1887/8, 379, 380, 381 note 1).

As an example of what he meant, Cantor once gave the equation 5 = 2 + 3. Two and three, he reasoned, are not contained in the concept five. Were they, he asked, what would that mean of one and four? 1, 2, 3, 4 are, however, virtual components of 5, he explains, and the equation indicates a specific ideal connection of the three cardinal numbers two, three and five for themselves. It is thus, he concludes, five in and for itself independent from four or three and from any other number. Each number is by essence a simple concept in which a manifold of ones are combined together into an organic whole in special ways (Cantor 1887/8, 418 note 1).

A good measure of the freedom Cantor felt he enjoyed in fact came from his adoption of the Platonic eidetikoi or ideal numbers alluded to in the above paragraph. And in affirming that the whole real numbers were "related to the arithmoi noetoi or eidetikoi of Plato with which they probably even fully coincide" (Cantor 1884, 84) and that his transfinite numbers were but a special form of these eidetikoi (Cantor 1887/8, 420; Cantor 1884, 84), Cantor himself provided an important clue as to how exactly he thought one might understand the seemingly paradoxical union of the One and the Many.

According to Zeller, Plato expressed the combining of the One and the Many by referring to Ideas as numbers, thus distinguishing between an empirical treatment of numbers and pure, ideal arithmoi eidetikoi which by their very nature are detached from things perceptible by the senses and which unlike the other, mathematical, numbers stand in a before and after relationship to one another. The essential thing for Plato, Zeller maintains, was only the thought, which underlies his number theory, that "in what is real the One and the Many must be organically combined" (Zeller 1875, 574). In such a theory, he explains, numbers become the connecting link between Ideas and appearance. It is by this combining of the One and Many that Plato was able to "put the concreteness of Socrates's concepts in the place of the abstract, Eleatic One, to link the concepts dialectically, and place them in not merely a negative, but also a positive relationship to appearance, that the Many of appearance is borne by and included in the unitary concept". And this, then, gave him the right to set forth a multiplicity (Vielheit) of logically interrelated Ideas, a world of Ideas (Zeller 1875, 583-84; 567-70; Zeller 1839, 239-41; see also Aristotle Metaphysics Books M-N).

According to Cantor's theories, the numbers obtained through his abstraction processes, both the whole numbers (cardinal numbers, powers) and the order types (the Anzahlen, or numbers of a well-ordered set) were simple conceptional formations, each of which was a genuine unity (monas) in which a plurality and manifold of ones became bound together in a uniform fashion. Abstracting from both the characteristics of the elements of the set and the order in which they are given, we obtain the cardinal numbers or powers; abstracting only from the characteristics of the elements and leaving their order intact, we obtain the ideal numbers or eidetikoi (1887/8, 379-80; 1883, 180-81).

The elements of a set, he explains, are to be thought of as separate. In the intellectual image (intellektualen Abbild), which he calls the order type or ideal number, the ones (Einsen) are, however, united into a single organism. In a certain sense, he explains, each ideal number can be looked upon as something composed of matter and form. The conceptually distinct ones contained therein supply the matter, while the order subsisting among them corresponds to the form (Cantor 1887/8, 380).

In finite sets, Cantor tells us, these two kinds of numbers coincide. The differences, however, come most distinctly to the fore in infinite sets (1887/8, 379-80). In the Mannigfaltigkeitslehre, he describes his real delight in seeing how when we proceed up into the infinite, the concept of whole number, which in the finite only serves as a backdrop for ideal numbers, splits into two concepts. And how in proceeding back down from the infinite to the finite he sees how beautifully and clearly the two concepts become one again and flow together into the concept of finite whole number. Without these two concepts, Cantor believed one could not progress in his theory of manifolds and he lamented Bernard Bolzano's failure to resort to this distinction (Cantor 1883, 180-81).

In 1890 Cantor wrote to Giuseppe Veronese that contradictions he had found in Cantor's theories were but apparent and that one must distinguish between the numbers which we are able to grasp in our limited ways and "numbers as they are in and for themselves, and in and for the Absolute intelligence", each of which "is a simple concept and a unity, just as much a unity as one itself. Taken absolutely", he explained, "the smaller numbers are only virtually contained in the bigger ones. They are, taken absolutely, all independent one from the other, all equally good and all equally necessary metaphysically" (Cantor 1991, 326). In 1895 he wrote to Charles Hermite that "the reality and absolute uniformity of the whole numbers seems to be much stronger than that of the world of sense. That this is so has a single and quite simple ground, namely the whole numbers both separately and in their actual infinite totality exist in that highest kind of reality as eternal ideas in the Divine Intellect" (cited Hallett 1984, 149).

Another clue as to how Cantor might realize a Platonic union of the One and the Many lies in Cantor's pronouncement that by a manifold or set he generally meant "any Many which can be thought of as a One, any totality of determinate objects which can be united by a law into a whole". He thus believed he was defining, he wrote, something related to the Platonic eidos or idea and to what Plato called a mikton in the Philebus, an ordered mixture of the peras (limit) and the apeiron (limitlessness, indeterminacy), what Cantor himself called the improper or potential infinite (Cantor, 1883, 204 note 1).

It was, in fact, in the Philebus, Zeller tells us, that the solution to the problem as to how unitary concepts might traffic in the multiplicity of appearance was said to lie in the principle that what is real (das Wirkliche) unites unity and plurality, limit and limitlessness (Zeller 1875, 567 note 1). It was there, Zeller explains, that Socrates showed that the One is Many and the Many is One, and that this was as true, not just of what is changing and transient, but of pure concepts which are also composed of the One and the Many, and have within them limit and limitlessness (Zeller 1875, 565).

In the dialogue Socrates confesses to having been perplexed by the assertion that the Many are One and the One is Many (14C-15D). The best way of avoiding chaos he had found, he says, lies in appealing to a method which had often eluded him and left him alone and confused, but by which every matter appropriate for scientific consideration had been brought to light and of which he had always been enamored. He said that superior people of old closer to the gods had received this method as a divine gift to mankind (16A-C).

According to those people of old, things said to exist consist of one and many and also have limit and limitlessness inherently within themselves. Being so composed it must be assumed that they have a single concept in every case. This must be looked for and, being present, will be found. Grasping it, one must find out whether in the next stage there are two or three or some other number. And each of these units must be treated in the same way until it is seen both that the original unity is One and Many and an unlimited number, and just how many it is. Only when the whole number between the unlimited and the single unit has been grasped can the concept of limitlessness be applied to the plurality, and only then can each of the units be dismissed and released into the indeterminate. Misguided people make the unit limitless right away and fail to demarcate what is intermediate (16C-17A).

By doing our best to assemble fragments and segments and recognize the mark of a single nature, everything found becoming more and less is to be classified as unlimited and put into a single class (24E-25A), which is then to be mixed with the class of limit, the class of whatever keeps opposites from from being at odds and makes them proportionate and harmonious by implanting number (25D). The mixing of these two classes, Socrates maintains, gives rise to a third class made possible by the measures which are produced with the limit, and from this, all fine things come (26B-D). As an example Socrates gave vocal sound which, he explains, is both a single phenomenon and quantitatively unlimited, but when one knows the nature and boundaries of the intervals with their height and depth of sound and systems that they form, the scales, tempos and measures does one become really knowlegeable. The other alternative, that of limitless plurality, he says, condemns one to ignorance (17B-D). (Compare with Cantor's examples of sets: Cantor 1887/8, 421-22; 412).

5. Husserl on Abstraction and the Concept of Number

Taught by Franz Brentano to despise metaphysical idealism, Edmund Husserl came within close range of Cantor's ideas upon arriving at the University of Halle in 1886 to prepare his Habilitationsschrift entitled On the Concept of Number (Husserl 1887). In this short work, a revised and much enlarged version of which was later published as the Philosophy of Arithmetic (Husserl 1891), Husserl set out to lay bare the roots out which arithmetic develops by logical necessity. Cantor served on Husserl's Habilitationskommittee and approved the mathematical portion of the work (Gerlach and Sepp 1994). The two men become close friends (M. Husserl 1988, 114; Cantor 1991).

On the Concept of Number bore the subtitle "Psychological Analyses" and in it Brentano's disciple set out to anchor arithmetical concepts in direct experience by analyzing the actual psychological processes to which, he believed, the concept of number owed its genesis. In view of his entire training, he later said, it had been obvious to him when he started the work that "what mattered most for a philosophy of mathematics was a radical analysis of the 'psychological origin' of the basic mathematical concepts" (Husserl 1913, 33). Psychology, he maintained in the introduction to the work, was the indispensible tool for analyzing the concept of number, and the analysis of elementary concepts one of the more essential tasks of the psychology of the time. "In truth", Husserl even declared there, "not only is psychology indispensable for the analysis of the concept of number, but rather this analysis even belongs within psychology" (Husserl 1887, 94-95).

Brentano tried inculcate in his students a model of philosophy based on the natural sciences and taught them to abhor the very kind of metaphysical idealism that pervaded Cantor's work so that any such considerations are markedly absent from both On the Concept of Number and Philosophy of Arithmetic. Yet, in spite of that major difference, many of the basic convictions underlying what Husserl hoped to accomplish in On the Concept of Number and the Philosophy of Arithmetic were perfectly compatible with the embattled creator of set theory's efforts to put the new numbers he was inventing on sound foundations. "With respect to the starting point and the germinal core of our developments toward the construction of a general arithmetic," Husserl wrote in about 1891, "we are in agreement with mathematicians that are among the most important and progressive ones of our times: above all with Weierstrass, but not less with Dedekind, Georg Cantor and many others" (Husserl 1994a, 1).

For one, it was Husserl's goal to analyze the concepts and relations which are in themselves simpler and logically prior, and then move on to analyze the more complicated and more derivative ones. The definitive removal of the real and imaginary difficulties in problems on the borderline between mathematics and philosophy, he then believed, would only come about in this way (Husserl 1887, 94-95).

And, under the influence of Weierstrass, Husserl still trusted that "a rigorous and thoroughgoing development of higher analysis... would have to emanate from elementary arithmetic alone, in which analysis is grounded", that the sole foundation of elementary arithmetic was "in that never-ending series of concepts which mathematicians call 'positive whole numbers'", and that "all the more complicated and artificial forms which are likewise called numbers --the fractional and irrational, and negative and complex numbers-- have their origin and basis in the elementary number concepts and their interrelations" (Husserl 1887, 95).

It was also Husserl's early conviction that set theory lay at the basis of mathematics. The most primitive concepts, Husserl informed his readers, are the general concepts of set and of number which are grounded in the concrete sets of specific objects of any kind whatsoever and to which particular numbers are assigned. Just as cardinal numbers relate to sets, he affirmed in the introduction to the Philosophy of Arithmetic, so ordinals relate to series which are themselves ordered sets (Husserl 1891, 4).

Citing Euclid's classical definition of the concept of number as "a multiplicity of units", eine Vielheit von Einheiten, Husserl began the analyses of the Philosophy of Arithmetic by asserting that "the analysis of the concept of number presupposes the concept of multiplicity (Vielheit)" (p. 8). He then comments that in place of the word 'Vielheit' the practically synonymous terms 'Mehrheit', 'Inbegriff', 'Aggregat', 'Sammlung', 'Menge', etc. have been used, and he informs his readers that in order to neutralize any differences in meaning among the terms, he will not restrict himself to the exclusive use of any one of them (p. 8 and note). (In the parts of the Philosophy of Arithmetic under study here, he uses 'Inbegriff', 'Menge', and 'Vielheit' interchangeably. Since Cantor also used the terms 'Menge', 'Mannigfaltigkeit' and 'Inbegriff' interchangeably, I have chosen to adopt the common term 'set' ('Menge'), the choice which I consider best limns the issues in any comparative study of Husserl's and Cantor's theories in the late twentieth century).

Husserl began his analysis of the concept of number affirming that there could be no doubt that multiplicities or sets of determinate objects were the concrete phenomena which formed the basis for the abstraction of the concepts in question. And he furthermore maintained that everyone knew what the terms 'multiplicity' and 'set' meant, that the concept itself was well-defined and that there was no doubt as to its extension which, though we may not as yet be entirely clear as concerns the essence and formation of the corresponding concept, might be taken as given (Husserl 1887, 96-97, 111; Husserl 1891, 9-10, 13).

Initially limiting himself to analyses of sets of individually given, collectively grouped objects and numbers as known through through direct experience (like Cantor's finger example) Husserl turned to the experience of concrete sets to consider how it is from them that both the more indeterminate universal concept of set and the determinate number concepts are to be abstracted (Husserl 1891, 10, 13).

And just as Cantor was trying to show that the transfinite numbers were "produced through abstraction from reality with the same necessity as the ordinary finite whole numbers by which alone all other mathematical conceptual formations thus far been produced" (Cantor 1991, 136), a theory of abstraction (which Husserl considered as psychological process) would be granted a significant role in On the Concept of Number (Husserl 1887, 97-98, 115-16), and even more so in Philosophy of Arithmetic (Husserl 1891, 10-16, 82-96, 165-66).

The concrete phenomena upon which abstraction is performed, Husserl told his readers, were sets of determinate objects of any kind whatsoever. Any objects "whether physical or mental, abstract or concrete, whether given in experience or fantasized", Husserl emphasized, can be grouped together in a set and be counted (Husserl 1891, 10). However, if the nature of the particular contents makes no difference at all in forming the general concept of set and definite number, then the question arises, Husserl realized, as to what sort of abstraction process might enable one to obtain the general concepts of set and numbers from such concrete groups of objects. For according to the traditional theory of concept formation through abstraction, one is to disregard the properties which distinguish the objects while focussing on the properties they have in common which then go into the making of the general concept. But, as Husserl explains:

It is right away obvious that a comparison of the individual contents we come across in the sets given would not yield the concept of set or number for us, and it was (even if this occurred) absurd to expect it…. The sets are merely composed of individual contents. So how are any common properties of the whole to come to the fore when the constituent parts may be so utterly heterogeneous? (Husserl 1891, 13; also Husserl 1887, 97, 112).

Such an objection may be overcome, Husserl explains, by observing that the sets are not merely composed of individual contents. Any talk of sets or multiplicities necessarily involves the combination of the individual elements into a whole, a unity containing the individual objects as parts. And though the combination involved may be very loose, there is a particular sort of unification there which would also have to have been noticed as such since the concept of set could never have arisen otherwise. "So", Husserl concludes, "if our view is correct, the concept of set arises through reflection on the particular... way in which the contents are united together... in a way analogous to the manner in which the concept of any other kind of whole arises through reflection upon the mode of combination peculiar to it". Husserl called the mode of combination characteristic of sets collective combination (Husserl 1891, 14-15; Husserl 1887, 97-98).

Husserl is now ready to characterize the distinctive abstraction process which produces the concept of set. Having before us any determinate, individual contents combined into a collection, he informs readers, we abstract the general concept by totally abstracting from the characteristics of the individual contents collected. Abstracting from something, he explains, simply means not paying any particular attention to it, and that absolutely does not cause the contents and their interconnections to disappear from our consciousness. Our interest is mainly focussed on the collective combination while the contents are only considered and attended to as any content whatsoever, anything whatsoever, any unit whatsoever (Husserl 1891, 84-85).

Number, the author of the Philosophy of Arithmetic explained, is the general form of a set under which the set of objects a, b, c falls. To obtain the concept of number of a concrete set of like objects, for example A, A, and A, one abstracts from the particular characteristics of the individual contents collected, only considering and retaining each one of them insofar as it is a something or a one, and thus regarding their collective combination, one obtains the general form of the set belonging to the set in question: one and one, etc. and... and one, to which a number name is assigned (Husserl 1891, 88, 165-66; Husserl 1887, 116-17).

Owing to confusions about the precise nature of the abstraction process Cantor and Husserl advocated, it is important to note that both men considered the abstraction process which yields the concept of set to be of a distinctive kind (Cantor 1991, 363, 365; Cantor 1887/8, 380, 411; Husserl 1891, 84). In the Philosophy of Arithmetic, Husserl underscored this conviction with criticisms aimed at dissociating the theory he adopted from the better known abstraction theories of Locke and Aristotle, something it is important to stress because philosophers have been all too wont to assimilate the process Husserl advocated to theories more familiar to them (Hill 1991, 15, 68). The analyses of the Philosophy of Arithmetic, Husserl declared, lead to the important, securely substantiated discovery that one cannot elucidate the formation of the concept of number in the same way one elucidates the concepts of color, form, etc. and that Aristotle and Locke were wrong to try to do so (Husserl 1891, 91-92). Mill was no more successful in winning Husserl's favor (Husserl 1891, 167).

In the late 1880s that is how Husserl believed that that most elementary of all arithmetical concepts, the concept of whole number could be produced through abstraction from reality to provide a sound basis for deriving more complicated and artificial forms of numbers in a strictly logical way. Change, however, was on the horizon and hints of the changes to come are already to be found in the Philosophy of Arithmetic. For conspicuously absent from that work is the enthusiastic espousal of psychologism of the opening pages of On the Concept of Number. And conspicuously apparent in the opening pages of the 1891 work is Husserl's tergiversation regarding his earlier conviction that all the more complicated and artificial forms of numbers had their origin and basis in the concept of positive whole numbers and their interrelations and could be derived from them in a strictly logical way. Weierstrass's thesis is never embraced in the Philosophy of Arithmetic in the enthusiastic way it was in On the Concept of Number. Instead we find Husserl initially writing that he will use it as his point of departure, but that it may prove false, and then finally in the introduction written just before the work was published, a statement that it is false (Hill 1991, 81-88).

6. Husserl Reasons his Way into the Realm of the Ideal

Completely under Brentano's influence in the beginning, Husserl had initially viewed idealistic systems with a jaundiced eye (Husserl 1919, 345). There were, however, ways in which Brentano's psychologism never came to satisfy him, and as he analyzed the basic concepts of mathematics he grew increasingly troubled by doubts of principle as to how to reconcile the objectivity of mathematics with psychological foundations for logic. His inability to silence his doubts undermined his confidence in psychologism, and he felt increasingly pushed to probe more deeply into essence of logic, and especially to try to resolve "the profound difficulties which are tied up with the opposition between the subjectivity of the act of knowledge and the objectivity of the content and object of knowledge (or of truth and being)" (Husserl 1994a, 250; Husserl 1900-01, 42).

Husserl repeatedly said that the immediate cause of his intellectual crisis lie in his inability to answer questions about "imaginary" numbers (ex. negative square roots, negative, irrational, complex, transfinite numbers) that arose while trying to complete the Philosophy of Arithmetic (Husserl 1913, 33; Husserl 1891, viii, 5-6; Husserl 1970, 430-47; Husserl 1994a, 15-16; Chapter 9 of this book). By 1891 we find him already protesting that "a utilization of symbols for scientific purposes, and with scientific success, is still not therefore a logical utilization" (Husserl 1994a, 48), and he lamented the mental energy wasted in "the endless controversies over negative and imaginary numbers, over the infinitely small and the infinitely large, over the paradoxes of divergent series, and so on" (Husserl 1994a, 49). How much quicker and more secure the progress of arithmetic would have been, he believed, "if already upon the development of its methods there had been clarity concerning their logical character" (Husserl 1994a, 49). He thought one might "search logical works in vain for light on what really makes such mechanical operations, with mere written characters or word signs, capable of vastly expanding our actual knowledge concerning the number concepts" (Husserl 1994a, 50). He knew of "no logic that would even do justice to the very possibility of a genuine calculational technique" (Husserl 1994a, 17).

Husserl has also told of how his mathematical investigations left him facing a host of burning questions about the incredibly strange realms of actual consciousness and of pure logic (Husserl 1994a, 490-91), a category comprising "all of the pure 'analytical' doctrines of mathematics (arithmetic, number theory, algebra, etc.) and the entire area of formal theories... the theory of manifolds in the broadest sense" (Husserl 1913, 28), "the traditional syllogistic... the pure theory of cardinal numbers, the pure theory of ordinal numbers, of Cantorian sets... the pure mathematical theory of probability" which it would be absurd to classify under psychology (Husserl 1994a, 250). He had no idea of how to bring the two worlds together. Yet he believed they had to interrelate and form an intrinsic unity (Husserl 1994a, 490-91). He wanted to know how symbolic thinking was possible, how objective, mathematical and logical relations constituted themselves in subjectivity, how insight into that was to be understood, and how the mathematical in itself, as given in the medium of the psychical, could be valid (Husserl 1913, 35).

All this ambiguity, Husserl said, "found ever new nourishment in the expanded philosophical-arithmetic studies, which extended to the broadest field of modern analysis and theory of manifolds and simultaneously to mathematical logic and to the entire sphere of logical in general" (Husserl 1913, 35). Logical research into formal arithmetic and the theory of manifolds presented him with particular difficulties (Husserl 1900-01, 41).

He confessed that he had been disturbed, and even tormented, by doubts about sets right from the very beginning. The concept of collection in Brentano's sense, Husserl explained, was to arise through reflection on the concept of collecting. Sets, he thus had reasoned, arose out of collective combination, in being conceived as one. This combining process involved when objects are brought together to make a whole only consists in that one thinks of them "together" and was obviously not grounded in the content of the disparate items collected into the set. It could not be physical, so it must be psychological, a unique kind of mental act connecting the contents of a whole. But then, he asked, echoing Cantor's concern to maintain the distinction between counting and numbers, was "the concept of number not something basically different from the concept of collecting which is all that can result from the reflection on acts?" (Husserl 1913, 34-35; re. Husserl 1887, 97-112, 115 and Husserl 1891, Chapter 3).

Husserl's questions were ones Cantor's philosophy of arithmetic raises. And if it seems exaggerated and unwarranted to suggest that someone's logical assumptions could be shaken by an encounter with Cantor's work, it is wise to remember that Husserl was on hand as Cantor began discovering the antinomies of set theory (Dauben 1979, 240-70), which played such a role in rocking the ground upon many others had hoped to derive arithmetic. Indeed it was in studying Cantor's 1891 proof by diagonal argument that there is no greatest cardinal number that Russell discovered the contradiction of the set of all sets that are not members of themselves (Grattan-Guinness 1978; Grattan-Guinness 1980; Hill 1991, 1), which David Hilbert described as having had "a downright catastrophic effect in the world of mathematics", having led Dedekind and Frege to abandon their theories and "quit the field". Hilbert characterized the reaction to Cantor's theory of transfinite numbers as having been "dramatic" and "violent" (Hilbert 1925, 375).

Cast intellectually adrift as his earliest convictions gave way, Husserl set off on his own to answer his questions (Husserl 1913, 17; Husserl, 1900-01, 41-43), ultimately reasoning his way into the realm of ideal as he began to accord idealist systems the highest value, seeing them as shedding light on totally new, radical dimensions of philosophical problems. The ultimate and highest goals of philosophy, Husserl came to believe are only opened up when the philosophical method these particular systems call for is clarified and developed (Husserl 1919, 345).

Although his experience of Cantor's work may have acted to pry Husserl away from psychologism and to steer him in the direction of idealism, Husserl said it was Hermann Lotze's work which was responsible for the fully conscious and radical turn from psychologism and the Platonism that came with it. In his Logic Lotze had unequivocally proclaimed that psychology's achievements, "do not reach those obscure regions of enquiry, the illumination of which might open new paths to Logic", and that he would "like to make clear... that Logic would have to renounce for a long time yet any profounder understanding of the operations of thought if she had to look for it in the psychological analysis of their origin" (Lotze 1888, § 333).

Husserl said that he gained his first major insight in studying Lotze's interpretation of Plato's doctrine of Ideas, which exercised a profound effect on him and became a determining factor in all his further studies. Lotze, Husserl realized, was already writing about truths in themselves, and so the idea came to him to transfer all of the mathematical and a major part of the traditionally logical into the realm of the ideal. His own concepts of "Ideal" significations, and "Ideal" contents of presentations and judgments originally derived from Lotze (Husserl 1913, 36; Husserl 1994a, 201; Lotze 1888, Chapter II).

Only by reflecting on Lotze's ideas did Husserl find the key to what he called "the curious conceptions of Bolzano" that had originally seemed naive and unintelligible to him (Husserl 1994a, 201). Husserl had known Bolzano's writings since he was a student of Weierstrass, had studied the 'paradoxes of infinite' under Brentano, and knew Bolzano's work through Cantor (Husserl 1913, 37), but had been disinclined to traffic in any mystico-metaphysical exploitation of Ideas or ideal possibilities and such (Husserl 1994b, 39). Though references to numbers in themselves and things in themselves appeared in Husserl's earliest writings (Hill 1991, 80-88), Husserl had originally thought of "propositions in themselves" as mythical entities, suspended between being and non-being (Husserl 1913, 37-38; Husserl 1994a, 201-02).

Viewed from the vantage point of Lotze's theory of Platonic Ideas, however, Bolzano's ideas came to have a powerful impact on Husserl. Plato, Lotze had insisted, never asserted the existence of the Ideas apart from things (Lotze 1888, § 317), but intended to teach what Lotze called their validity (die Geltung), "a form of reality not including Being or Existence". In relegating the Ideas to a home which is not in space, Lotze argued, Plato was not trying to hypostasize their mere "validity" into any kind of real existence, but was plainly seeking to guard altogether against any such attempt (Lotze 1888, §§ 314, 317, 318, 319).

It suddenly became clear to Husserl that Bolzano had not hypostasized presentations and propositions in themselves (Husserl 1994b, 39), which Husserl now saw

were what were ordinarily called, the 'senses' of statements, what is said to be one and the same when, for example, different persons are said to have asserted the same thing, or what scientists simply call a theorem (for example the theorem about the sum of the angles in a triangle) which no one would think of as being someone's experience of judging and that this identical sense could be none other than the universal, the species belonging to a certain Moment present in all actual assertions having the same sense which makes possible the identification in question, even when the descriptive content of individual experiences of asserting varies considerably in other respects (Husserl 1994a, 201).

"Viewed this way", Husserl echoed Lotze, "Bolzano's theory that propositions are objects which nonetheless have no 'existence' is quite intelligible. They enjoy the ideal existence or validity characteristic of universals (Husserl 1994a, 201-02).

The first two volumes of the Wissenschaftslehre on presentations and propositions in themselves, Husserl concluded, were to be seen as a initial attempt to provide a unified presentation of the domain of pure ideal doctrines and already provided a complete plan of a pure logic, an insight which proved immensely helpful enabling him to use Bolzano's account step by step to verify the "Platonic" interpretation (Husserl 1913, 37). Lying hidden in Bolzano's Wissenschaftslehre was something that seemed to Husserl to be "one of the most momentous logical insights" that the "core content of any normative and practical logic consists in propositions that do not deal with acts of thought, but rather with those Ideas instanced in certain of their Moments" (Husserl 1994a, 209), a key thesis of Lotze (Lotze 1888, Chapter II).

However, neither Bolzano nor Lotze provided the full answers Husserl needed for there was no trace in the work of either of any thought of the pure phenomenological elucidation of knowledge that Husserl ultimately concluded was needed to solve the puzzles surrounding the being in itself of the ideal sphere and its relationship to consciousness (Husserl 1994b, 39; Husserl 1913, 38-40, 45-49; Husserl 1994a, 201-03, 209. Husserl 1900-1, Prolegomena, § 61, Appendix). These could only be solved through his own phenomenological investigations for which logic had to be more than formal, mathematical, theory, but required "phenomenological and epistemological elucidations in virtue of which we not merely are completely certain of the validity of its concepts and theories, but also truly understand them" (Husserl 1994a, 215). These phenomenological investigations became the stuff of the Logical Investigations (Husserl 1900-01) that Husserl wrote during his last decade in Halle and in which he has said that all possible efforts had been "taken to dispose the reader to the recognition of this ideal sphere of being and knowledge... to side with 'the ideal in this truly Platonistic sense', 'to declare oneself for idealism' with the author" (Husserl 1913, 20).

7.Concluding Remark

By describing the crucial shift in Husserl's views on psychologism and metaphysical idealism as they relate to Georg Cantor's theories of abstraction and Platonic Ideas, I have tried to fit these two major themes of Cantor's philosophy of arithmetic into a now rather standard version of the development of Husserl's thought. I have used textual analysis to achieve my goals. Plumbing Husserl's and Cantor's writings of the 1880s and 1890s, I have described something of the relationship between the two men's efforts to provide solid foundations for the positive whole numbers and the more complicated and artificial forms of numbers, thereby establishing connections between their ideas which have until now gone virtually unsuspected. In so doing, I hope to have contributed to filling a gaping hole in the understanding of the development of the ideas which went into the making of phenomenology, as well as to a better understanding of Cantor's philosophy of arithmetic by examining the philosophical, metaphysical theories within which, to the chagrin of his contemporaries, he chose to frame his ideas.

It is my conviction that further research into the relationship between Husserl's and Cantor's ideas will show that no complete grasp of Husserl's phenomenology, and his philosophy of logic and mathematics in particular, is possible without a clear understanding of the role Cantor's philosophy of arithmetic played in the development of Husserl's thought. And I would like to take this opportunity to suggest that a thorough understanding of the impact of Cantor's ideas on all periods of Husserl's work will prove fruitful and constructive in ways which, given then the current state of research into the subject, could only seem surrealistic at the present time. Cantor's work embodied many the very problems Husserl found so distressing, and it is certain that in searching for solutions he felt compelled to turn elsewhere. However, further research will surely show that phenomenology, while not being Cantorian, distinctly bears Cantor's imprint.

Acknowledgments

I wish to express my gratitude to Ivor Grattan-Guinness for all the advice and documents he has given me, to Miss Emiko Ima for her hard work, and to William Gallagher for help transliterating the Greek terms.

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