Cantor's concept of set in the light of Plato's Philebus.
IN THE OPENING SENTENCE of his final mathematical publication, Beitrage zur Begrundung der transfiniten Mengenlehre, Cantor states in a few words what h.e means by a set.By a "set" we understand every collection M of definite well-distinguished objects m of our intuition or our thinking (called the "elements" of M) into a whole.'
Here, however, we shall focus on an earlier "definition" that intimates concerns about the ontological status of collections. It appeared in Cantor's Grundlagen einer allgemeinen Mannichfaltigkeitslehre (2) which summed up the quintessence of his deepest contribution to mathematics--the transfinite numbers--both from a mathematical and a philosophical point of view. In an end note Cantor emphasizes that Mannichfaltigkeitslehre is to be understood in a much more encompassing sense than that of the theory of sets of numbers and sets of points he had been developing up to that stage.
For by a "manifold" or "set" I understand in general every Many which may be thought of as a One, i.e., every totality [Inbegriff] of determinate elements that can be united by a law into a whole.
He then goes on to explicate this further by drawing upon Plato's Philebus.
And with this I believe to define something related to the Platonic eidos or idea, as well as that which Plato in his dialogue "Philebus or the Supreme Good" calls mikton. He contraposits this against the apeiron, i.e., the unlimited, indeterminate which I call inauthentic-infinite, as well as against the peras, i.e., the limit, and he declares the [mikton] an orderly "mixture" of the latter two. That these two notions are of Pythagorean origin is indicated by Plato himself; cf. A. Boeckh, Philolaos des Pythagoreers Lehren. Berlin 1819. (3)
The aim of this article is to shed some light on these two passages in Grundlagen. Specifically, I propose a substantive connection between Plato's mature theory of ideas and an interpretation of Cantor's 1883 definition that, arguably, is in accordance with the meaning of "set" in contemporary set theory. Among other things this will allow a construal of the term "law" that does not entail any restriction to definable sets.
For the most part, the perspective from which Cantor's thought will be presented here is that of Grundlagen. In particular, this means that the ordinals are conceived of as numbers "constructed from below" by the principles of generation described in [section] 11 of that work in contrast to the "purely mathematical" treatment of ordinals as canonical representatives of order types of well-ordered sets adopted by Cantor in all of his subsequent publications. (4) I also incorporate notions which, although they found their explicit expression at later stages, were undoubtedly known to the author of Grundlagen. (5) Similarly, ideas from contemporary set theory such as large cardinals are discussed. They are not explicitly present in Cantor's writings but nevertheless can be traced back to the fundamental intuitions underlying Cantorian set theory. The question in what sense Cantor's 1883 definition differs from the one of 1895 as well as other aspects of the latter will be investigated on another occasion.
Our interpretation of Plato relies, to the extent possible, on the systematic exposition by Eduard Zeller. (6) The intention, however, is not to vindicate this author against numerous studies that have been undertaken since then, but to use a source that Cantor evidently regarded as congenial to his own reading of Plato. (7)
I
Set, Eidos and Idea. On ontological grounds, Cantor's comparison of sets with the Platonic eidos or idea is thoroughly intelligible. Either one of those notions may be conceived of as a One over Many insofar as they denote what is common to a plurality of individuals of the same name. (8) For Plato, this One over Many is an autonomous entity just as, for Cantor, a set is distinct from the multiplicity of its elements.
However, in the first instance Cantor thought of sets as particulars rather than universals. Some indication for this is provided by his repeated references to a set as a "thing for itself," which can then occur as an element in another set, in distinction to the "general concept (universal)" of its cardinality or order type. (9) This does not rule out the linkage with Plato though, for in Plato's ontology, arguably, ideas can be construed as individual entities rather than general concepts. Cantor himself equates Plato's arithmoi eidetikoi with Gedankendinge (objects of thought) in his introduction of ordinals as order types. (10) Furthermore, Cantor's formulation "every Many that may be thought of as a One" (11) suggests that for him sets are abstract particulars, that is, entities belonging to the realm of ideas.
Nevertheless, Cantor maintains that certain sets, for example, the first few number classes, possess physical manifestations. (12) Indeed, his declared goal is
to study the relationships of transfinite numbers not only mathematically, but to document and investigate them in all places where they occur in nature. (13)
The rationale for such an investigation, as pointed out by Cantor himself, goes back to Plato which means there is a certain analogy between the epistemic roles performed by ideas and by sets in the systems of these two thinkers. According to Plato ideas hold the key to understanding the world of the senses: meaning and truth can be found in the existence of individual things only to the extent that the latter partake in the ideas. (14) In a similar vein Cantor envisaged his theory of sets to furnish an "organic explanation of nature" that is superior to a "mechanical" one. (15)
This peculiar trait in Cantor's mathematical realism finds its clearest expression in his distinction of the intrasubjective (or immanent) and transsubjective (or transient) reality of ideal entities. The former refers to the reality of concepts and ideas "insofar as they occupy, by virtue of definitions, a specific place in our thought, are strictly distinguished from all other constituents of our thinking, stand in determinate relationships to them and thereby modify the substance of our minds in a determinate manner." The latter kind of reality is ascribed to ideas "which must be regarded as an expression or an image of processes and relationships in the external world that confronts the intellect." (16) Cantor sees himself in essential agreement Plato when he invokes the Eleatic One to argue that those two realities are intrinsically linked to each other. (17)
Undoubtedly Cantor's primary motivation for these remarks as well as the ensuing conception of "free mathematics" (18) is to defend the objective existence of actually infinite sets. Rather than pursuing this any further we will delve directly into Plato's Philebus for gaining insights into Cantor's concept of set and its relevance for contemporary set theory.
II
Apeiron, Peras, and Mikton in the Philebus. At first sight Cantor's recourse to Plato's Philebus seems farfetched. After all, the leading theme of that dialogue is a question in ethics. Should the good life be devoted to wisdom or be driven by pleasure? (19) Embedded in the text, however, is another problem which is intimately related to Cantor's definition: the principle "that One should be Many or Many One" (20) and its ostensible violation of the Eleatic doctrine that the absolute being of the One does not admit a state of change or division. This becomes an issue already at the beginning of the dialogue in the interlocutors' attempt to arrive at a definition of pleasure which is One "yet surely takes the most varied forms." (21)
From a modern perspective, unity is related to how a whole unifies its parts or a subject its attributes. Those dichotomies, however, have no place in Plato's thinking. His difficulties begin with unities which do not belong to the class of things that are born and perish. Do such unities--denoted by generic terms like man, beauty, the good have a real existence? And how can they maintain their identity and simultaneously be dispersed and multiplied in the infinity of the world of generation? (22) According to Plato, the answers to these questions must be devised by way of a true dialectic rather than skilful disputation (eristike). The latter is what gave rise, already in the Phaido, (23) to paradoxical questions about the composite nature of natural persons, questions that Plato here dismisses as "childish and detrimental to the true course of thought." (24) Plato's dialectician, by contrast, breaks down a complex unity into smaller ones employing schemes of classification into genera and species. (25) This introduces an aspect of measure and quantity that is interpolated, as it were, between unity and infinity. (26)
In this manner Plato attempts to resolve a central issue for his theory of ideas, the question how the inherent multiplicity among the ideas can be integrated into an intelligible whole. In the subsequent discussion, however, the Many is portrayed in a way that seems to apply exclusively to our sensory experiences. Here the previously employed strategy of classification into genera and species must fail. For in the material world, things always appear to us with qualities admitting of more or less. This is the conception of infinity referred to as apeiron in the Philebus: everything that lacks determination and boundary and is capable of infinite variation. (27) From that Plato distinguishes a second form of being which "is at rest and has ceased to progress." (28) Banishing the more or less, it allows instead for "their opposites, that is to say, first of all equality, and the equal, or again the double, or any other ratio of number and measure." (29) The epithet used by Plato to characterize this sphere of existence is peras which is also the Greek word for limit or boundary.
The novel idea in the Philebus is that an orderly interaction of the peras with the apeiron brings about stable formations in the realm of sense perception which are capable of maintaining their identity amidst the continuously changing variation of their elements. (30) Plato uses the term mikton for such formations and characterizes their mode of existence as a "birth into true being" (31) or, alternatively, as "being which has emerged." (32) Both of these terms indicate that in a mikton we have a mixture of the two ontological spheres of Becoming (the sensory world) and Being (ideal forms).
To illustrate this concept Plato remarks that the spoken "sound which passes through the lips whether of an individual or of all men is one and yet infinite." (33) Note that our ability to discriminate unambiguously between myriads of spoken sounds is conditioned on the availability of an ideal system, that is, the phonetic system of the language. Although the physical utterances of individual speakers correspond only approximately to the phonemes, the law-like structure of the phonetic system enables determinate demarcations. A graphic example for this is provided by the spoken sound of a "th" in English. In German the same sound is classified as a mispronounced "s." Thus, borrowing Plato's formulation, "the knowledge of the number and nature of sounds is what makes a man a grammarian." (34)
In sum, the general scheme found in all examples of a mikton provided by Plato is this: definite delineations in the realm of becoming, to the extent that they are possible at all, rely on certain ideal structures that supervene (35) on the sensory perceptions. In this way with each mikton an area of knowledge is associated: meteorology with the seasons, medicine with health (or disease), and harmonics with musical sounds. Moreover, for Plato, the cause of the intertwining of peras and apeiron is to be found in reason. This also clarifies the relation in which reason stands to the good life, for the latter is itself a mikton, brought into being through imposing the limit of law and order on unbounded pleasure and self-indulgence. (36)
III
Apeiron, Peras, and Mikton in Cantor. When one tries to set these notions in relation to Cantor in a straightforward way, various difficulties ensue. One option is to compare Cantorian sets and the Platonic mikton from the epistemological point of view. Cantor, as we have seen, conceived of set theory as an investigation into the essential structures of natural reality. (37) In a similar manner the mikton in the Philebus is indispensable for gaining knowledge about the material world. However, in view of the terseness and cryptic wording of Cantor's remarks one can only speculate about the exact role of set theory in his envisaged "organic explanation of nature." (38) Moreover, the motivations underlying this enterprise do not bear any resemblance to the concerns of modern day set theorists. Finally it needs to be pointed out that the stipulation in Cantor's 1883 definition that a set is a "totality of definite elements" has no equivalent in the mikton when the latter is regarded primarily as an epistemological device for reaching a systematic understanding of natural phenomena. Apparently, what Cantor meant by "definite" is that the potential elements of a set must be well-distinguished as a precondition for membership. (39) By contrast, in the Philebus, definite demarcations in the sensory world are effected by a mikton.
In the case of apeiron, matters are complicated by Cantor's translation of the term as "inauthentic infinite" in his 1883 definition of "set." On the first page of Grundlagen we are told that "inauthentic infinite" denotes a "variable, but always finite quantity." (40) Thus one may be tempted to conclude that in Cantor's 1883 definition apeiron is synonymous with the indeterminacy residing in the potential infinity of the integers. However, under this narrow construal of apeiron, the linkage of Cantor's concept of set with the Philebus remains elusive for two reasons. First, it is inconceivable what the unfinished character and "borrowed reality" by which Cantor characterizes the inauthentic infinite would contribute to the true existence of a set which is a "finished thing for itself." (41) Second, if Cantor were to translate the Platonic apeiron into "inauthentic infinite," he would turn the ontological order in the Philebus on its head given his refusal in Grundlagen to ascribe any form of existence to the inauthentic infinite. (42) For in the Philebus the apeiron is understood as an absence of boundary or demarcation which, however, does not result from a removal of boundaries, but rather precedes their introduction. (43) Thus, in this sense, the apeiron is ontologically prior to the peras.
A more enlightening reading of apeiron in Cantor can be extracted from a letter to A. Eulenburg dated February 28, 1886 where Cantor lays out the philosophical ideas underlying his distinctions of various notions of infinity. After repeating essentially the definition in Grundlagen of "inauthentic infinite," which he now calls "potential infinite (apeiron)," (14) he continues
More generally, I speak of a potential infinite in all places where an indefinite quantity is considered which is capable of innumerably many determinations. (45)
The two principles of generation in [section] 11 of the Grundlagen are one way of producing "innumerably many determinations" as their repeated application allows one "to overcome any limit in the generation of concepts of real integer numbers." (46) This is apparent already from the initial segment of the transfinite numbers given as follows. (47)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It may be objected that [[epsilon].sub.0], the least [epsilon]-number, (48) is a closure point for the numbers generated in this manner from [omega]. Nevertheless, Cantor's contention that in the formation of transfinite numbers via the first two generation principles "we seem to get lost in the unlimited" (49) is essentially correct. The reason is that one can readily name a "definite succession of defined real whole numbers" (50) converging to [[epsilon].sub.0], and, by virtue of the second generation principle, [[epsilon].sub.0] can then be added as the next element in the progression of transfinite numbers. In a similar fashion one may overcome closure points of uncountable cofinality such as the uncountable cardinals [[ALEPH].sub.n] with n [greater than or equal to] 1, for Cantor's second generation principle does not entail a restriction to successions of length [omega]. (51) The basic idea is that, given a closure point [gamma], the set of its predecessors in their canonical enumeration yields a definite succession of numbers to which the second generation principle may be applied in order to obtain [gamma] as a new number. (52) Thus, in a first approximation, we may construe the apeiron as the open-endedness in the generation of transfinite numbers.
Another dimension of the indeterminacy inherent in this generation becomes apparent upon closer inspection of the above scheme. Each of the operations in the scheme naturally gives rise to another operation transcending it: addition is superseded by multiplication, multiplication by exponentiation, exponentiation by its unbounded iterations into the transfinite. A similar phenomenon occurs, over and over again, in the higher regions of infinity: every principle postulating the existence of "large" sets turns out to be the initial step in a natural hierarchy of ever stronger principles. From inaccessible cardinals one is led to Mahlo cardinals, from the Mahlo hierarchy to the hierarchy of indescribable cardinals, then (via some extrapolations) to measurable cardinals, the latter serving themselves as the starting point of a hierarchy of still larger cardinals. (53) As a general rule, for each cardinal K of a particular type there are already K many cardinals of each weaker type below K. In fact, much stronger degrees of transcendence obtain.
To be sure strong axioms of infinity and the orders of transcendence charting the higher infinite were unknown to Cantor. Nevertheless, the basic intuition underlying their formation is implicit in Cantor's doctrine of the Absolute maintaining the uncharacterizability of the totality of all sets in its open-endedness. (54) Some inkling of this is provided by the absolutely unlimited growth of the sequence of transfinite numbers which Cantor regarded as an "appropriate symbol of the Absolute." (55)
Moving beyond the generation of transfinite numbers to the formation of sets, we may conceive of the apeiron in Cantor as a general reference to the inexhaustibility of the operation "set of." This interpretation accords well with Cantor's conception of a general theory of manifolds aiming at the widest possible mathematical theory of collections. That universality, in effect, was already adumbrated by Bernhard Riemann who contended that any collection whatsoever of given things determines a "discrete manifold." (56) More recently Kurt Godel, in his famous article on Cantor's continuum problem, stressed the universality and transcendence of the concept of set.
The operation "set of x's" (where the variable "x" ranges over some
given kind of objects) cannot be defined satisfactorily (at least
not in the present state of knowledge), but can only be paraphrased
by other expressions involving again the concept of set, such as
"multitude of x's," "combination of any number of x's," " part of
the totality of x's," where a "multitude" ("combination," "part")
is conceived of as something which exists in itself no matter
whether we can define it in a finite number of words (so that
random sets are not excluded). (57)
Again, there are no explicit expressions of this wider construal in Cantor's writings. However, anecdotal evidence that Cantor did think along these lines comes from his usage of the metaphor of an "abyss" in describing his conception of "set" to F. Bernstein. It is worthwhile quoting Bernstein's recollection in its entirety.
With regard to the concept of set Dedekind remarked: he envisaged a set in the manner of a closed sack containing determinate things which, however, one does not see and of which one knows nothing, except that they exist and are determinate. A short time later Cantor revealed his conception of set: he erected his colossal figure up high, raised his arm describing a magnificent gesture, and spoke with a glance directed to the indeterminate: "A set I imagine to be like an abyss. "(58)
That this is indeed an appropriate characterization becomes apparent already by considering the power set of the smallest infinite ordinal. Modern work in set theory has shown that even in this seemingly restricted setting, the operation "set of" is capable of generating very high orders of transcendence. (59) Another graphic illustration is provided by the remarkable history of Cantor's continuum problem. (60) Its many unexpected turns and especially the striking results that have been obtained recently reinforce the perception that one is looking down into an abyss whose bottom appears ever more distant each time one gets a better view.
The peras displays itself most directly in Grundlagen by way of the Hemmungs- oder Beschrankungsprinzip (principle of limitation) "through which certain limits are successively imposed upon the thoroughly endless generation process" (61) of transfinite ordinals allowing thereby quantitative comparisons. Thus, strictly speaking, the Hemmungsprinzip is not a method for generating numbers but rather a device for demarcating canonical halting points in the endless progression of transfinite numbers. (62) It may be argued that the transfinite numbers themselves are manifestations of peras since they represent "fixed boundaries" being "completed infmities." (63) Finally at the most fundamental level, an aspect of peras is at work in Cantor's stipulation that the elements of a set already possess some internal articulation in so far as they are clearly discernible individuals. (64)
From a modern point of view, the peras manifests itself in each of the classification schemes that have been devised for delineating the totality of all sets. Among other things, this includes the stratification of the universe into the cumulative hierarchy of von Neuman ranks, the fundamental set-construction principle of [member of] -induction, and also the linear ordering of strong axioms of infinity according to their consistency strength. (65) The latter is of course an outgrowth of the number classes which Cantor obtained from the Hemmungsprinzip. The former two ultimately rest on Cantor's stipulation that a set consists of previously given elements, that is, his adumbration of what is now called the "axiom of foundation." (66)
With these remarks the connection of Cantor's concept of set with the Platonic mikton becomes more transparent. On the one hand, a "general theory of manifolds" aiming to account for all possible collections is confronted with the inexhaustibility of the operation "set-of." On the other hand, as a mathematical enterprise, Cantorian set theory seeks to establish definite demarcations in the open-ended totality of all sets. This is achieved by means of mathematical tools like the ones mentioned in the preceding two paragraphs. (67) All of them reveal determinate structures which supervene on the apeiron that is inherent in the operation "set of," thus comprising "laws"--to use Cantor's own formulation--for uniting a Many into a One. In other words, sets belong to Plato's third class of existence whose members are the offspring of apeiron and peras. The combination of their elements into a whole, as in the case of a mikton, is "effected by the measure which the limit introduces." (68)
Clearly this does not entail a restriction to definable sets. The proof of the well-ordering theorem is a case in point. Here the equivalent of what Cantor in his 1883 definition calls a "law" is given by a transfinite recursion employing the axiom of choice, but this establishes the existence of a well-ordering without exhibiting a definition. (69)
To see that Cantor did not intend any restriction to definable sets, one only has to consider his remarks on the "internal determinateness" of the question whether two sets are equinumerous. (70) From the discussion it is clear that the "law" by which elements of the two sets are put in correspondence need not be definable. Another indication is provided by the frequent occurrences of the term "law" in Cantor's statements of various theorems. Invariably, the proofs of those theorems make use of a "law" only to the extent that one is dealing with a fixed but otherwise unspecified mathematical entity, that is, a sequence of numbers, a function, or a set, whose definability plays no role in the argument. (71)
To summarize, our analysis of the linkage of Cantor's concept of set with the Platonic mikton exploited three interrelated aspects: set theory as a scientific discipline, principles of set formation, and sets as a specific ontological category. While it may be objected that this blurs important distinctions, from a methodological point of view it does resemble the threefold explication in the Philebus. There the mikton is simultaneously described in terms of a field of knowledge (for example, the phonetic system, harmonics, meteorology), as a principle or form of existence (birth into true being), and as a specific kind of entity (the spoken sound, a musical note, the seasons).
Under this interpretation, set theory assumes the form of a dialectic which already manifests itself in the restricted setting of transfinite numbers in the opposition between the first two principles of generation and the principle of limitation. Interestingly, Cantor himself does not use the term dialectic on that occasion. However, in connection with the canonical progression of Unendlichkeitssymbole--the prototypes of transfinite numbers serving as indices for transfinite iterations of the derivative of a point set--Cantor expressly speaks of a "dialectical generation of concepts." (72) That should be compared with the aforementioned characterization of the dialectic method in the Philebus: "the infinite must not be suffered to approach the many until the entire number of the species intermediate between unity and infinity has been discovered." (73)
In addition the dialectic element is also present in the notion of the transfinite itself where Plato's two forms of existence, being and becoming, are combined. For the transfinite, as emphasized by Cantor, truly exists as "completed infinity," (74) yet at the same time it retains an aspect of the characteristic mark of the apeirion--the more or less--in the sense that "its abundance of forms and features point with necessity to an... absolute maximum" whose size can neither be increased nor decreased. (75)
IV
Pythagorean Roots. With regard to the Pythagorean origin of apeiron, peras, and mikton, Cantor is correct on two accounts. First, in the Philebus Plato refers to a tradition, "handed down by the ancients who were our betters and nearer the gods than we are." According to that tradition
Whatever things are said to be are composed of one and many, and have the finite and infinite implanted in them. (76)
Second, the first fragment attributed to Philolaus declares that "in the world order nature was assembled out of unbounded and boundary-forming components, both the world order as a whole and all things that are in nature." (77) At the beginning of the second fragment we are told:
With necessity, all things that are must either be forming boundaries or be without boundary or both at the same time. (78)
Despite these similarities, the question remains whether and to what extent the preserved texts are an authentic expression of Pythagorean thought. Cantor ostensibly accepted the view of the influential 19th century philologist August Boekh who regarded these fragments, "with the exception of a few pieces by Archytas, as the most authentic relics of the Pythagorean school." (79) That must have been congenial to Cantor's own interests in presenting his transfinite numbers and their arithmetic as a natural extension of the integers. Moreover, the centrality of the number concept in Cantor's world view (80) meant that he had to be troubled by the paradox of the One and the Many. For the concerns that are expressed in the Philebus, like the objections raised in the Parmenides, (81) not only create difficulties for Plato's theory of ideas but, in effect, threaten the very basis of the Pythagorean system by exposing a fundamental disharmony in the concept of number and discrete plurality.
Leaving aside the question about the authenticity of the fragments there is, however, another aspect under which we may compare Philolaus' ontological scheme with the one in Philebus. Notice that the former, although he makes a universal statement, merely speaks of individual "things that are" and there is no mention ever of apeiron or peras as abstract principles. Indeed from Homeric times through the age of Ionian philosophy the apeiron was invariably associated with matter. In etymological terms it originally denoted what is visually experienced in the perception of something which has no boundary, for example, the sea or the sky. (82) Plato took the bold step of eliminating any material connotation from the apeiron construing it instead as the universal principle that is at work in every creation of more or less. (83)
But according to Aristotle, who conceived of matter as pure potentiality (dynamis), Plato did identify the apeiron with matter that is common to physical things and to (ideal) numbers or ideas. Zeller argued that the identification--presumably an attempt to explain how the ideas bring about the existence of physical objects--amounts to a blatant misunderstanding of Plato. (84) This also seems to have been Cantor's view since he cites with approval the corresponding section in Zeller while expressing a preference for Plato's philosophy of the infinite over Aristotle's. (85) Certainly one factor for this preference was Aristotle's denial of the actual existence of quantitative infinity. Moreover for Cantor, who regarded (infinite) sets as "finished things," a reading of Plato that "transfers the apeiron, understood in the same sense in which it marks the peculiar trait of existence in the sensory world, to the ideas" (86) would clearly be unacceptable.
There is, however, at least one other factor which, moreover, underlines the significance of the Philebus for Cantor from a different perspective. Recall that Aristotle, too, acknowledges two components in the spoken word: the uttered sound (phone') and something nonphysical (logos) which lends meaning to it. (87) The latter roughly corresponds to the ideal sound system of the Philebus that supervenes on the apeiron of sound utterances. (88) However, for Aristotle the relationship between phone and logos is one of form (eidos) to matter (hyle), in analogy to the manner in which the eidos as individual essence, say of Pallas Athena, is indispensible for obtaining a statue from a lump of bronze. By contrast, the abstract law-like structures in the Philebus giving rise to a mikton are decidedly not forms expressed in matter, but rather they are Uberformungen (supervening frameworks) that are invariant against all matter. (89) In a similar vein, Cantor gave a new meaning to the apeiron, free from any material connotations, construing it as the inexhaustibility of the operation "set of x's" where x can be anything whatsoever. (90)
Technische Universitat Berlin, Institut fur Mathematik
Acknowledgements: Research partially funded by MEC Grants HUM2006-05029 and HA2006-0108, and by the John Templeton Foundation. I am indebted to my colleagues M. Reeken and L. Tengelyi for stimulating discussions.
(1) Georg Cantor, "Beitrage zur Begrundung der transfiniten Mengenlehre," Mathematische Annalen 46 (1895): 481; Georg Cantor Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo (Berlin: Julius Springer, 1932), 282. All translations are my own unless otherwise noted.
(2) Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten Nr. 5," Mathematische Annalen 21 (1883): 545-86; Ges. Abhandlungen. 165-209. Separate publication as Georg Cantor, Grandlagen einer allgemeinen Mannichfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (Leipzig: Teubner, 1883). Henceforth abbreviated as Grundlagen.
(3) Ges. Abhandlungen, 204, end note 1.
(4) Ges. Abhandlungen, 195-7. The first principle of generation allows one to pass from a given number a to its successor [alpha] + 1. The second principle of generation allows one to add as a new number the least upper bound of "any definite succession of defined whole real numbers" having no last element. It needs to be emphasized that "generation" must not be understood in the sense that numbers are "created" by these principles, a view that would be incompatible with the brand of Platonism advocated in Grundlagen, [section] 8. Rather, what is generated through these principles is an awareness of transfinite ordinal numbers whence they become objects of cognition (see Ges. Abhandlungen, end note 6, 207). Possible reasons for Cantor's switch from the generative concept of ordinal numbers--in spite of its philosophical advantage (for example, in connection with the paradoxes) of highlighting the open-endedness inherent in the progression of transfinite numbers--to the "purely mathematical" treatment of order types of well-orders may be gleaned from his letters to Mittag-Leffler of September 23, 1883 and to Labwitz of February 15, 1884, see Georg Cantor Briefe, ed. H. Meschkowski and W. Nilson (Berlin: Springer, 1991), 130, 178, as well as from the opening pages of his "Principien einer Theorie der Ordnungstypen. Erste Mitteilung" in "An Unpublished Paper by Georg Cantor," ed. I. Grattan-Guiness, Acta Mathematica 124 (1970): 65-107. A well-order R on a set M is a total order of its elements such that any non-empty subset of M possesses a frost element in the sense of R. Two well-orders are isomorphic if there is a one-to-one correspondence between their underlying sets respecting the order, that is, if x precedes y under the first well-order, the same hold for their corresponding images with respect to the other well-order. The isomorphism relation defines an equivalence relation on the totality of all well-orders. The ordinal numbers serve as canonical representatives of the equivalence classes, that is, for every well-order R there is a (unique) ordinal a such that the well-order given by the predecessors of a in their natural enumeration is isomorphic to R.
(5) One example is the power set axiom which states that for any set M, its power set (the set of all subsets of M) exists. It is first mentioned by Cantor in a letter to Hilbert in 1898 (Briefe, 396) although the underlying intuition had a formative influence on the development of set theory from its very beginning (for example, the power set axiom is needed to guarantee the existence uncountable transfinite numbers). Notice that in Cantor's letter to Hilbert, the power set axiom is presented as a corollary to a "definition" of set which is reminiscent of the one in Grundlagen (modulo the preoccupation of the former with warding off inconsistent multiplicities).
(6) Eduard Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung. 2.1. Sokrates und die Sokratiker, Plato und die alte Akademie, 3. Auflage (Leipzig: Fues, 1875).
(7) Ges. Abhandlungen, 205, end note 2 and 206, end note 6.
(8) Republic 10.596a; see Zeller, Die Philosophie der Griechen, 553-4, 562.
(9) Ges. Abhandlungen, 379, 411, 416, 420, 422; Briefe, 146. See also his letters to Dedekind, Ges. Abhandlungen, Appendix and his letters to Hilbert, Briefe, 390, 393-4, 396, 426, 440 where, however, the main focus is on distinguishing sets from "inconsistent multiplicities."
(10) Georg Cantor, "Principien einer Theorie der Ordnungstypen. Erste Mitteilung," in I. Grattan-Guiness, "An Unpublished Paper by Georg Cantor," Acta Mathematica 124 (1970): 84.
(11) Ges. Abhandlungen, 204, end note 1 (emphasis added).
(12) For the purpose of illustration see the hypothesis about the cardinalities of the set of corporeal monads and the set of ether monads put forward in the closing remarks of Georg Cantor, "Uber verschiedene Theoreme aus der Theorie der Punktmengen in einem n-fach ausgedehnten, stetigen Raume [G.sub.n]. Zweite Mitteilung," Acta Mathematica 7 (1885); see Ges. Abhandlungen, 275-76.
(13) Ges. Abhandlungen, 205, end note 2.
(14) See. Ges. Abhandlungen, 206, end note 6. In addition to the passage from Zeller, Die Philosophie der Griechen, 541 cited by Cantor see also Zeller, Die Philosophie der Griechen, 544-5, 547.
(15) Letter to Wundt of March 4 1883, Briefe, 135. In a letter to Mittag Leffler dated September 22 1884, Cantor states that "a more penetrating investigation into the nature of all things organic" had been occupying his mind for the past 14 years and indeed provided the original motivation for "the tedious [...] business of studying point sets." Briefe, 202. 16 Ges. Abhandlungen , 181.
(17) "But to the same degree as our presentations (Vorstellungen) possess truth this presupposition Plato shares with others (Parmenides)--their objects must possess reality, and vice versa. What can be recognized, exists, what cannot be recognized does not exist, and to the same extent that something exists, it is also recognizable." Zeller, Die Philosophie der Griechen, 541-2 as cited in Grundlagen, 206-7, end note 6. In addition Cantor appeals to Spinoza (compare footnote 42) and to Leibniz.
(18) Ges. Abhandlungen , 182-3.
(19) Plato, Philebus 11a-12b. Quotations from the Philebus are taken from Benjamin Jowett, The Dialogues of Plato translated into English with Analyses and Introductions in Five Volumes, 3rd edition revised and corrected (Oxford: Oxford University Press, 1892).
(20) Plato, Philebus 14c.
(21) Ibid., 12c-d.
(22) Ibid., 15a-c. Recall the example at Phaedo 102b of Simmias who possesses both tallness and smallness at the same time being taller than Socrates and smaller than Phaedo.
(24) Plato, Philebus 14d-e.
(25) Ibid. 16d-17a.
(26) Ibid. 17d-e.
(27) Ibid. 24a-25a.
(28) Ibid. 24d.
(29) Ibid. 25a-b.
(30) Ibid. 25e-26b
(31) Ibid. 26d.
(32) Ibid. 27b.
(33) Ibid., 17b.
(34) Ibid., 17b, emphasis added.
(35) Throughout this paper "supervene" is used not in the technical sense of analytic philosophy but in the sense of coming or occurring as something additional, extraneous, or unexpected.
(36) Plato, Philebus 26b.
(37) In addition to the references in footnote 15 see Ges. Abhandlungen, 235 and Briefe, 65-6, 134-5, 142, 202-3, 224-5, 227, 228-9, 231, 255, 285, 292-3, 298-9, 314, 426, 459.
(38) One hint to what Cantor may have had in mind is provided in his letter to Wundt of Oct. 16, 1883, Briefe, 142. There he claims to have established "with strict rigor" the countable infinity of the "totality of organic cells in the cosmos." Presumably, this alludes to the fact that there cannot be an uncountable family of pairwise disjoint nonempty intervals in Euclidean space. See Georg Cantor "Uber unendliche, lineare Punktmannichfaltigkeiten Nr. 3," Mathematische Annalen 20 (1882): 117; Ges. Abhandlungen, 153. Another clue comes from his demonstration of the possibility of continuous movements in discontinuous spaces, Ges. Abhandlungen, 154-7. Evidently, Cantor hoped to employ the conceptual resources of point set topology for the vindication of the theories of Boscovich, Cauchy, Ampere and others concerning the "constitution of matter." Briefe, 224-5, 228-9. For Cantor, the point of contact with those thinkers--whose ideas laid the ground for the concept of field in modem physics--is the opposition to the emerging atomism in physics and chemistry of those days. He preferred instead to construe the ultimate constituents of matter along the lines of Leibniz' Monadology (see Ges. Abhandlungen, 275-6) whose metaphysics he regarded "infinitely many times closer to the truth" than that of Newton. Briefe, 292. Indeed Cantor was convinced that his quantitative theory of infinity holds the key to the resolution of certain difficulties in the metaphysical systems of Leibniz and Spinoza, Ges. Abhandlungen, 177.
(39) Compare Cantor's explanation of what it means for a set to be "well-defined" in "Uber unendliche, lineare Punktmannichfaltigkeiten Nr. 3," Mathematische Annalen 20 (1882): 114-15; Ges. Abhandlungen, 150. The requirement that the members of a set need to be "well-distinguished" also figures prominently in Cantor's definition of 'set' in Beitrage, 481; see also Ges. Abhandlungen, 379, 411, 420. The partial overlaps among the subsets of a given set seem to be a potential worry in Cantor's stipulation of the power set axiom Briefe, 396.
(40) Ges. Abhandlungen, 165 (emphasis in original); see also 166 ("something that is variable but finite"). This specific definition of 'inauthentic infinite' also appears in places other than Grundlagen. For example, in his letter to Wundt of October 5 1883, Cantor explicitly identifies the "variable number which remains finite" with the inauthentic infinite. Briefe, 139.
(41) Cantor uses the scholastic epithet "synkategorimatice infinitum" to express that the apeiron, construed as inauthentic or potential infinite, merely has a "borrowed reality," and he contrasts this with the "kategorematice infinitum = finished infinitum" given by a transfinite set. Georg Cantor, letter to Eulenburg of Feb. 28 1886, Ges. Abhandlungen, 404; see also the letter to Labwitz of Feb. 15 1884, Ges. Abhandlungen, 391. The characterization of a set as a "finished thing for itself' is a recurring theme in Cantor's discussions of sets vs. inconsistent multiplicities; see the letters to Dedekind and Hilbert cited in footnote 9.
(42) "To the indeterminate, variable, inauthentic infinite, in whatever form they may appear, I cannot ascribe being, for they are nothing but either relational concepts or purely subjective presentations resp. intuitions (imaginationes), in no case adequate ideas." Ges. Abhandlungen, 205, end note 3. The contrast between "imaginationes" and "adequate ideas" is reminiscent of Spinoza who, in Part II of the Ethics, distinguished between imprecise and confused sensory images on the one hand and adequate ideas on the other. The latter are formed in a rational and orderly manner, and in so far as they are considered in themselves, without relation to the object, have all the properties or intrinsic marks of a true idea. Ethics, part II, definition 4. As pointed out in Ges. Abhandlungen, 206, end note 5, Spinoza's definition of "adequate" coincides with the meaning of "immanent reality of concepts" in Grundlagen [section]8.
(43) Plato, Philebus 25e, 26c, 27b.
(44) In an earlier letter to G. Enestrom dating from November 4, 1885 Cantor states expressly that "potential infinite" is synonymous with "inauthentic infinite" as introduced in Grundlagen; see Ges. Abhandlungen, 376.
(45) Ges. Abhandlungen, 401.
(46) Ges. Abhandlungen, 197
(47) Here [omega] denotes the smallest transfinite number which Cantor conceived of as the least upper bound of the progression of integers, as expressed in the second principle of generation. Nowadays, the existence of [omega] is stipulated axiomatically (that is, by the axiom of infinity). Cantor, however, was led to the transfinite numbers in a different way. During his studies of the uniqueness of representations of functions by trigonometric series (see Roger Cooke, "Uniqueness of Trigonometric Series and Descriptive Set Theory, 1870-1985," Archive for History of Exact Sciences 45 (1993): 281-334) he discovered the operation of forming the derivative of a given subset of the real numbers. (The derivative of a set A is the set of all its cluster points, that is, all points for which each neighborhood contains infinitely many points from A). Clearly this process can be iterated any finite number of times producing a nested sequence of sets. Now the key point is that there is a canonical choice for the [omega]-th iterate, namely the intersection of all the n-fold iterates. At a) the counting process may be continued yielding transfinite numbers [omega] + n for each integer n via repeated application of the first principle of generation (Initially, in this context Cantor spoke of symbols of infinity [Unendlichkeitssymbole] indexing the infinite series of derivatives of a point set rather than full blown numbers. The switch to numbers occurred once he conceived of the possibility of a transfinite arithmetic for these symbols of infinity; see his letter to Mittag-Leffler of Oct. 25 1882, Briefe, 91). The least upper bound of all the [omega] + n is denoted by [omega] + [omega] which equals [omega] . 2 in transfinite arithmetic. Iterating this process [omega] times one obtains the numbers co * n for each integer n. Their least upper bound is denoted by [omega] . [omega] or [[omega].sup.2]. Iterating multiplication with co yields the numbers [[omega].sup.n] for each integer n. Their least upper bound is denoted by [[omega].sup.[omega]], and by further iteration one obtains a stack of n powers of co for each integer n.
(48) The least [epsilon]-number is a stack of height [omega] of powers of [omega], that is, it is the least upper bound of all the stacks of height n of powers of [omega]. Generally, [epsilon]-numbers are the fixed points of the operation [alpha] [right arrow] [[omega].sup.[alpha]], that is, those [alpha] for which [[omega].sup.[alpha]] = [alpha]. For more information on the mathematical background of this article the reader may consult an introductory textbook of set theory such as Azriel Levy, Basic Set Theory (Berlin: Springer, 1979); Thomas Jech, Set Theory (New York: Academic Press, 2003); or Kenneth Kunen, Set Theory. An Introduction to Independence Proofs (Amsterdam: North Holland, 1980). The first appearance of [epsilon]-numbers is in Georg Cantor, "Beitrage zur Begrundung der transfiniten Mengenlehre," Mathematische Annalen 49 (1897): [section]20; Ges. Abhandlungen, 347-51.
(49) Ges. Abhandlungen, 196.
(50) Ibid. The "definite succession of defined integer real numbers" leading up to [[epsilon].sub.0] is defined recursively as [[alpha].sub.0] = [omega] and [[alpha].sub.n + 1] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for n [greater than or equal to] 1.
(51) With such a restriction Cantor would have been unable to progress beyond the second number class. The cofinality of a is the smallest ordinal [beta] such that there is a sequence of length [beta] of ordinals less than [alpha] with limit [alpha].
(52) This may seem a rather roundabout way. Note, however, that contrary to the modern point of view according to which each ordinal is identical with the set of its predecessors, in the ontological scheme of Grundlagen, numbers are not sets. To obtain the set of predecessors of [[ALEPH].sub.n + 1] the power set of the set of predecessors of [[ALEPH].sub.n] n in combination with replacement (formulated only 1898 in a letter to Hilbert Briefe, 396 but adumbrated in Grundlagen, see Ges. Abhandlungen, 200) and the isomorphism principle suffices. By the latter, which is implicit in Grundlagen, see Ges. Abhandlungen 170, any well-ordered set is isomorphic to some ordinal. In the case of inaccessible cardinals (that is, regular fixed points of the [ALEPH] function, see next footnote) and their generalizations appeals to Cantor's Absolute (see below) in the form of reflection principles are needed. However, inaccessible and other large cardinals obtained through reflection principles which state that for any property of the totality of all sets expressible in the language of set theory there exists a set having that property--are decidedly not generated "from below."
(53) A cardinal K is (strongly) inaccessible if it is closed under the power set operation and under forming sums of length less than K of smaller cardinals. Clearly, under this definition, [omega] is inaccessible. [The power set of a finite set is finite, and finite sums of finite numbers are finite.] The existence of uncountable cardinals which are inaccessible is a strong axiom of infinity. Mahlo cardinals are obtained by systematically thinning out the class of inaccessible cardinals in a similar way as Cantor iterated the formation of derivatives. A cardinal K is indescribable, if any attempt to characterize K by a formula of set theory (admitting higher types) fails in the sense that there is already a smaller cardinal satisfying the formula. Indescribable cardinals are of very high Mahlo order. Measurable cardinals were discovered in the wake of attempts to extend the Lebesgue measure to all sets of reals and are thus founded on entirely different ideas. Nevertheless, they can be related to the aforementioned cardinals by showing that they transcend them in an essential way. For a comprehensive introduction to strong axioms of infinity see Akihiro Kanamori, The Higher Infinite, 2nd edition (Heidelberg, Berlin, New York: Springer, 2003).
(54) Grundlagen, 175, 205; Ges. Abhandlungen, 378, 391, 405. Limitations of space prevent examining this central notion in Cantor's thought in any detail. A more extensive discussion can be found in Lazlo Tengelyi, "On Absolute Infinity in Cantor" (lecture manuscript, Universitat de Barcelona 2007), and in Kai Hauser, "Cantor's Absolute in Metaphysics and Mathematics", manuscript, Technische Universitat Berlin, 2009). The reflection principles mentioned in footnote 52 are deducible from Cantor's Absolute in various ways. For example, if the Absolute is to encompass all possibilities, then any set theoretic property P of the totality of all sets, V, must already hold of some set in this totality. Otherwise set formation would have to include V itself as a set, and the latter could not have been the totality of all sets to begin with. If P expresses closure under specific set forming operations, this immediately yields the existence of inaccessible and Mahlo cardinals. By reflecting reflection one obtains indescribable cardinals.
(55) Grundlagen, 205, end note 2. It is worth noting that the inherent open endedness in the generation of transfinite numbers expressly prohibits the application of the second generation principle to the totality of all numbers. In other words, the so-called antinomy of Burali-Forte can never arise if the transfinite numbers are generated "from below" as outlined in Grundlagen.
(56) Bernhard Riemann, "giber die Hypothesen, welche der Geometrie zu Grunde liegen", Habilitationsvortrag 1854, Abhandlungen der Koniglichen Gesellschaft der Wissenschaften Gottingen 13 (1867). Also in Gesammelte Mathematische Werke, wissenschaftlicher Nachlass und Nachtrage, 3. Band, ed. Raghavan Narasimhan, (Berlin, Leipzig: Springer, Teubner, 1990): 273-4.
(57) Kurt Godel, "What is Cantor's Continuum Problem?" in ed. P. Benacerraf and H. Putnam, Philosophy of Mathematics: Selected Readings (Cambridge: Cambridge University Press, 1964), 258-73; reprinted in Kurt Godel Collected Works, Volume II, ed. S. Feferman et. al. (Oxford University Press, New York, 1990): 259, footnote 14.
(58) Richard Dedekind Gesammelte Mathematische Werke, 3. Band, ed. Emmy Noether (Braunschweig: Fried, Viehweg and Sohn, 1932), 449. In Grundlagen the word "abyss" occurs only once in a polemic passage mocking Kronecker's finitism as an overreaction to the alleged danger of falling into the "abyss of the transfinite." Ges. Abhandlungen, 173.
(59) For example, certain sets of integers called sharps and daggers encode truth in inner models of set theory containing all ordinals as well as large cardinals of prescribed orders; see Ronald B. Jensen, "Inner Models and Large Cardinals," Bulletin of Symbolic Logic 1 (1995): 393-407. In a different direction, assuming the generalized continuum hypothesis, Jensen has produced "virtual" extensions of the universe (via class forcing) where a single set of integers reconstructs the entire universe. Aaron Beller, Ronald B. Jensen and Philip Welch, Coding the Universe, London Mathematical Society Lecture Note Series #47 (Cambridge: Cambridge University Press, 1982). We mean here the problem in its original formulation in Georg Cantor,
(60) "Ein Beitrag zur Mannigfaltigkeitslehre," Journal fur die reine und angewandte Mathematik (Crelles Journal) 84 (1878): 257; Ges. Abhandlungen 132; see also Cantor's remarks on the "internal determinateness" of mathematical questions in "Uber unendliche, lineare Punktmannichfaltigkeiten Nr. 3," Mathematische Annalen 20 (1882): 115-16; Ges. Abhandlungen, 150-1. For the latest developments see W. Hugh Woodin, "The Continuum Hypothesis," Notices of the American Mathematical Society 48 (2001): 567-76 and 681-90.
(61) Ges. Abhandlungen, 167. Curiously, the order of the principles of generation reflects the aforementioned primacy of the apeiron in Plato's ontological scheme where the introduction of limit and measure is subsequent to indeterminate existence (see Section 3)
(62) Letter to Wundt dated October 5 1883, Briefe, 138.
(63) Georg Cantor, Letter to Labwitz of Feb. 15 1884; Ges. Abhandlungen, 391.
(64) Georg Cantor "Uber unendliche, lineare Punktmannichfaltigkeiten Nr. 3," Mathematische Annalen 20 (1882): 114-15; Ges. Abhandlungen, 150.
(65) See the references in footnote 48 and Akihiro Kanamori, The Higher Infinite, 2nd edition (Heidelberg, Berlin, New York: Springer, 2003). The levels of the cumulative hierarchy are defined by transfinite recursion along the ordinals: [V.sub.0] = 0, [V.sub.a + 1] = P([V.sub.a]) and [V.sub.[lambda]] = [[union].sub.[alpha]] < [lambda] [V.sub.[alpha]] for limit ordinals [lambda]. By the axiom of foundation (see next footnote), any set occurs in one of these levels. The von Neuman rank of a set x is the smallest ordinal a such that x [member of] [V.sub.[alpha] + 1]. Propositions of the form "for all sets x, P(x) holds" may now be proved by induction on the von Neuman rank of x, that is, by [member of] -induction, in a similar fashion as one proves statements of the form "for all integers n, P(n) holds" by induction on the size of n. For strong axioms of infinity A and B, axiom A has higher consistency strength than axiom B, if the consistency of the axioms of set theory with A implies the consistency of the axioms of set theory with B and, moreover, this implication (via a suitable coding) is provable in number theory. Obviously this defines a partial ordering on the class of all strong axioms of infinity. The astonishing fact--in view of the ostensible disparity in the principles leading to strong axioms of infinity--is that (in the cases examined so far) this ordering is total, that is, any two strong axioms of infinity are comparable.
(66) The axiom of foundation states that every nonempty set has some member that is minimal under the membership relation. For an interesting account of how the cumulative hierarchy along with the associated iterative concept of set can be developed with tools that were available to the author of Grundlagen. See Ignacio Jane, "The Iterative Conception of Sets from a Cantorian Perspective," Logic Methodology and Philosophy of Science. Proceedings of the Twelfth International Congress, ed. Petr Hajek et al. (London: King's College Publications, 2005), 373-93.
(67) The list of devices should also contain model theoretic methods, forcing, inner models and descriptive-set theoretic concepts. It is only through the infusion of these sophisticated techniques that set theory is able to measure the combinatorial content of strong axioms of infinity.
(68) Plato, Philebus 26d. What this measure ensures is also freedom from contradiction. See Cantor's letter to Grace Chisholm-Young of March 9, 1907 where he writes--with explicit reference to his 1883 definition--that "set' denotes only those multiplicities "that can be thought of as units, that is, things, without contradiction." Herbert Meschkowski, "Zwei unveroffentlichte Briefe Georg Cantors," Der Mathematikunterricht 17, no. 4 (1971): 31.
(69) The axiom of choice states that for any partition of a given set into disjoint, nonempty subsets, there exists a choice set containing exactly one element from each of those subsets. The point of this axiom is that it is a pure existence principle in the sense that a choice set is not assumed to be definable. In fact, modem work in set theory suggests that the existence of definable choice sets for the continuum is highly problematic.
(70) Georg Cantor "Ober unendliche, lineare Punktmannichfaltigkeiten Nr. 3," athematische Annalen 20 (1882): 116; Ges. Abhandlungen, 151.
(71) In this connection it is also instructive to recall Godel's remark on "random sets" in the passage cited above.
(72) Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten Nr. 2," Mathematische Annalen 17 (1880): 358; Ges. Abhandlungen: 148; compare Georg Cantor "Ober unendliche, lineare Punktmannichfaltigkeiten Nr. 3," Mathematische Annalen 20 (1882): 114; Ges. Abhandlungen, 149.
(73) Plato, Philebus 16d-e. On a speculative note, we hazard that Cantor may be expressing that he was led to the discovery of a classification scheme for infinite totalities in a similar way as Plato's dialectic managed to break through the logical straight-jacket of the law of identity, namely through investigating comparative relationships among concepts. In the Philebus (see Section 3 of this article), these relationships--entailment versus exclusion, compatibility versus incompatibility--reestablished a systematic unity in the multiplicity inherent in the ideas. See Zeller, Die Philosophie der Griechen, 564-7. Cantor's classification scheme for infinite totalities relies on comparative relationships of size.
(74) Georg Cantor, Letter to Labwitz of Feb. 15 1884, Ges. Abhandlungen, 391.
(75) Georg Cantor, Letter to Eulenburg of Feb. 28 1886, Ges. Abhandlungen, 405. To P. Ignatius Jeiler Cantor wrote on Pentecost of 1888 that "every transfinite leads from the most varied sides directly to the Absolute, that is, with necessity it permits to deduce the existence of the Absolute by a dialectical inference through reason." Johannes Bendiek, "Ein Brief Georg Cantors an P. Ignatius Jeiler OFM," Franziskaner Studien, Jahrgang 47, Heft 1 (1965): 71. Note also that Zeller intimates that the combination of the One and the Many found in the ideas may have led Plato to conceive of ideas as numbers. Zeller, Die Philosophie der Griechen, 567.
(76) Plato, Philebus 16c-d.
(77) Hermann Diels und Walther Kranz, Die Fragmente der Vorsokratiker, achte Auflage, herausgegeben von Walther Kranz, erster Band (Berlin: Weidemannsche Verlagsbuchhandlung, 1956), Fragment 1.
(78) Hermann Diels und Walther Kranz, Die Fragmente der Vorsokratiker, Fragment 2.
(79) August Boeckh, Philolaos des Pythagoreers Lehren nebst den Bruchstucken seines Werkes (Berlin: Vossische Buchhandlung, 1819), 39. This assessment was called into question by Erich Frank who argued that the fragments were authored in their entirety by Plato's nephew Speusippus on the grounds that they already contain some of the leading ideas from the Timaeus. See Platon und die sogenannten Pythagoreer. Ein Kapitel aus der Geschichte des griechischen Geistes (Halle: Niemeyer, 1923). Walter Burkert by contrast, considers Plato's tripartite classification of apeiron, peras and mikton to be of genuine Pythagorean origin. See Weisheit und Wissenschaft. Studien zu Pythagoras, Philolaos und Platon (Nurnberg: Carl, 1962).
(80) See for example his letter to Hermite of November 30 1895. See Herbert Meschkowski, Probleme des Unendlichen, (Braunschweig: Friedrich Vieweg and Sohn, 1967), 262.
(81) Plato, Parmenides 127e, 128d.
(82) Wolfgang Schadewaldt, Die Anfange der Philosophie bei den Griechen (Frankfurt: Suhrkamp, 1978), 237.
(83) Plato, Philebus 24c.
(84) Zeller, Die Philosophie der Griechen, 628--39. The point of departure of Zeller's argument is Metaphysics 1.6.987b 18--20.
(85) Grundlagen, 205.
(86) Zeller, Die Philosophie der Griechen, 632.
(87) Aristotle, On the Generation of Animals 5.7.786b 21.
(88) Plato, Philebus 18b-d.
(89) Plato, Philebus 24e-25e.
(90) Modulo the stipulation that the collection of these x forms a "consistent multiplicity" of determinate elements. Briefe, 399, 407.
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| Author: | Hauser, Kai |
|---|---|
| Publication: | The Review of Metaphysics |
| Article Type: | Critical essay |
| Geographic Code: | 4EUGE |
| Date: | Jun 1, 2010 |
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