Georg Cantor:
The Next Century

Fidelio, Vol. III, No, 3. Fall 1994

Afterword by Lyndon H. LaRouche, Jr.
July 30, 1994

Georg Cantor: The Next Century

The relatively brief period of Halle-to-Rome correspondence between mathematical genius Georg Cantor and Cardinal Johann Baptiste Franzelin, S.J. remains one of the more significant anomalies in the history of science, and also theology. To appreciate the central feature of that correspondence itself, it is essential to identify some crucially relevant features of Cantor’s life: then, and during the decade following the termination of that exchange of letters.

Georg Cantor’s 1897 Contributions To The Founding of The Theory of Transfinite Numbers (Beiträge)¹ is an indispensable work; but, there are problems. Cardinal Johann Baptiste Franzelin’s abrupt termination, on Jan. 26, 1886, of his ongoing correspondence with Cantor,² is crucial for understanding fully the darkened mood which distinguishes Cantor’s writings of the 1890’s from those of the 1880’s; and that latter period in Cantor’s life is one of the keys to understanding the circumstances in which the correspondence was terminated.

Directly to the crucial issue: Cantor’s depression confronts the informed reader immediately at the outset of reading the Beiträge. Exactly as it is placed there in the 1962 edition,³ the evidence is:

“Hypotheses non fingo.” [—Newton].

That reference would not have been allowed by the Cantor of the Franzelin correspondence, the 1883-84 Grundlagen,⁴ or even the 1887-1888 “Mitteilungen zur Lehre vom Transfiniten.”⁵ The Cantor of 1897 and later, pleading for recognition from Britain, and engaging himself in such pathetic enterprises as the myth of Francis Bacon’s authorship of Shakespeare’s works,⁶ is no longer the Cantor of the 1880’s.

This mid-1890’s change in Cantor’s mood has been misused by sundry sophists as a pretext for deriding not only the 1897 Beiträge as “pathological science,” but also such earlier writings as the Grundlagen. There are problematic features in the Beiträge, but none to which those critics might rightly object. From the vantage-point of those who have studied the more vigorous Cantor writings of the 1880’s, the failing of the Beiträge is its propitiatory quality, its excessive reliance upon formalism, just as the dedicatory note to Newton might imply.

Since our purpose here is to situate the Cantor-Franzelin correspondence, we are permitted and obliged to dispense with the subsidiary mathematical formalities of the matter as much as possible. Under those circumstances, the immediately following descriptive observation is supplied.

All of the crucial conceptions met in the Beiträge are met in earlier writings of the 1883-1888 interval; the significance of the 1897 book is that it supplies a proof and some further elaboration of those conceptions from a strictly formal standpoint. The Georg Cantor of 1897, a mere fifty-two years of age, has become, in one very important sense of the term, “an old man,” his enemies have finally succeeded in quenching his creative spark. He is left to no more than commenting faithfully upon the achievements of a brilliant past state of mind, to which he is fated never fully to return. The operative term there is “reporting faithfully”; the discovery reported in the 1897 book is authentically Cantor’s, but, sadly, the exposition is added by a Cantor who could no longer make new such original discoveries.

If one takes all the relevant elements of Cantor’s environment into account, Cardinal Franzelin’s abrupt termination of the correspondence was at least a contributing cause for Cantor’s very-premature old age. The Cardinal clearly did not intend such an effect; the problem was, that the topics of that correspondence are the same issues which mobilized the rogues of the European science community, especially the mathematicians, in a two-decades-long aversive behavioral modification of Cantor. Those topics, which are the essential content of the correspondence, are the issues prompting Leopold Kronecker and his positivist accomplices to conduct one of the most widespread and disgusting inquisitions in the internal history of science, the virtual lynching of Georg Cantor.

Georg Cantor’s Theology

Georg Cantor, born of Jewish ancestry in St. Petersburg, Russia on March 3, 1845, began life with a grand heritage. He was the maternal grandnephew of the Joseph Boehm who was, in turn, the collaborator of Ludwig van Beethoven in the performance of Beethoven’s late string quartets, who was the founder of the Vienna school of violin performance, and personally the teacher of the famed violinist Joachim. That musical tradition permeated the family; until his adolescent turn into mathematics, Georg Cantor himself was trained as a violinist in this tradition, and two of his siblings, in addition to other immediate relatives, were notable musicians. The family converted to a Protestant rite, and moved to Germany, where he studied in such locations as Wiesbaden and Darmstadt.

During 1885-1886, this Jewish-born German Protestant, and music-student turned mathematical genius, is exchanging correspondence on some of the most profound issues of theology with an influential Cardinal in the Rome of Pope Leo XIII. To cap those ironies, Cantor was by no means unprepared.

This correspondence was prompted, on Cantor’s part, by a question addressed to him, asking whether he had seen a certain writing by French Abbot Francois Napoleon Marie Moigno.⁷ This provoked a Nov. 4, 1885 letter to one G. Enerstroem in Stockholm,⁸ and the enclosure of a copy of that letter in Cantor’s letter of Dec. 17, 1885 to Franzelin.⁹ The Cardinal acknowledged this communication in a letter of Dec. 25, 1885, cautiously rebuking Cantor’s criticism of Cauchy and Moigno with the suggestion that Cantor might abstain from the appearance of pantheism.¹⁰ To this, Cantor replied on Jan. 22, 1886. The response from the Cardinal was issued on Jan. 26, 1886, excusing himself from further correspondence with Cantor.¹¹ Cantor sent a “thank you” letter for consideration given on Jan. 29, 1886, but received no acknowledgement.¹²

To assess the Cardinal’s manifest reaction to Cantor’s attack on the characteristically neo-Aristotelian (e.g., positivist) fallacies of Cauchy and Moigno, one must take into account the reputation already gained in professional circles at that time by Cantor’s 1883-1884 Grundlagen.¹³ This work had mobilized Cantor’s enemies into attack at full tilt, led, as always, by Kronecker. Cantor’s reaction to the query respecting Moigno’s piece, is visibly a response to the already ongoing political lynch-mob being mobilized against him, in Germany, France, and elsewhere.

With the Grundlagen’s appearance, it is evident that he is well-grounded in Plato’s work, and is attempting to view the method of Leibniz from that standpoint. He has also shown himself a follower of Cardinal Nicolaus of Cusa in these matters. The appearance of the “Mitteilungen”¹⁴ affirms that continuing commitment. This establishes Cantor’s scientific and theological outlook very clearly for anyone with the prerequisites to assess this.

Briefly: Cantor himself insists that his science and theology center around two crucial points of equivalence between his own work on the transfinite and Plato’s principle of hypothesis. His opinion on these parallels is broadly correct.¹⁵ Cantor insists that his general notion of the Transfinite is equivalent to Plato’s Becoming, and that his own Absolute corresponds to Plato’s Good. By Becoming is signified Plato’s generalized notion of what Plato terms hypothesizing the higher hypothesis.¹⁶ Obviously, to follow the argument in Cantor’s letters (or, elsewhere, for that matter) one must first understand what is signified by Plato’s principle of hypothesis.

For the purposes of formal criticism, especially formal mathematics or mathematical physics, Plato’s principle of hypothesis is best presented in terms of his Parmenides: the ontological paradox of the One and the Many. His solution for that paradox is the formal definition of human creativity, as valid axiomatic revolutions in formal mathematical physics typify creativity, in the sense of Cantor’s definition of type. In Plato, the term hypothesis signifies such a type of discovery, and never anything different. Briefly, work through an illustration of Plato’s discovery of the principle of hypothesis.

The secondary student’s classroom model of reference for a Many is Euclid’s geometry: an expandable lattice-work of theorems, each and all mutually consistent with one another in terms of a shared, fixed set of axioms and postulates. That expandable list of theorems constitutes a Many. The challenge is to identify a single conception such that, when we think about that single conception, we are implicitly defining each and every theorem which might possibly be part of that theorem-latticework. If one adheres to the formalist methods of a Parmenides, a Sophist, an Aristotle, a Galileo, a Newton, a Cauchy, a Kronecker, a Bertrand Russell, or a John Von Neumann, no true solution to this ontological paradox is possible.¹⁷

However, let us discover a proposition which is true in nature, but which cannot be consistently a theorem of that theorem-latticework; let us designate that latter as theorem-lattice “A.” This theorem requires us to alter some part of the set of axioms and postulates of theorem-lattice “A” to the effect that all of the old theorems must now be scrapped in their earlier form, and recalculated on the basis of a new set of axioms and postulates, theorem-lattice “B.” In another case, nature obliges us to proceed to a third theorem-lattice, “C.” On this basis, Plato hints in writing the Parmenides, a solution for discovery of the One is attainable.

Instead of focussing upon fixed objects, such as sense-objects, one must focus upon change itself as the primary fact of nature, and of mental life. In the given case, it is the change from A to B, and from B to C, which is crucial. It is this change which one can conceptualize as an unified object of thought, a One. This permits us to conceptualize the changes in the respective underlying sets of axioms and postulates, from A to B, as a unit, as a One.

That One is an hypothesis. Any valid axiomatic-revolutionary discovery of that type is an instance of hypothesis as Plato defines hypothesis.

Next, continue with the illustration provided. Examine the successive changes, from A to B, B to C, and, then, C to D. This sequence of changes—of hypotheses—is a Many, too. Scrutiny of this Many enables us to conceptualize a higher sort of One. As the first level of One—e.g., A to B—defined an hypothesis, the new One required is a method of generating hypotheses: a higher hypothesis. It is a method of discovery. In natural science historically, there is evidence of various types of relatively valid methods of discovery, but some proving more valid than others. Study of the Many alternative, relatively valid choices of methods of hypothesis (higher hypotheses) yields Plato’s hypothesizing the higher hypothesis.

That latter, hypothesizing the higher hypothesis, is Plato’s knowledge of the Becoming. The notion of a One corresponding to a Many is Cantor’s notion of a transfinite; he is occupied with examining the general hierarchy of transfinitenesses as a domain defined in the sense indicated by Plato’s principle of hypothesis.

This principle of hypothesis implies the necessary existence of the Good. Since hypothesis is development in physical space-time, a Many, what is the One which corresponds to hypothesizing the higher hypothesis respecting physical space-time? It must be intelligence; it must be all space, all time, combined with efficient (creative) intelligence as One. That is Plato’s Good; that us what Cantor signifies by Absolute.

On this issue, the London-aligned political party within European science was united in a maenad’s hateful frenzy, not only against Cantor’s notion of the mathematical transfinite, but also the related work of Karl Weierstrass, Riemann, et al. earlier. This is a continuation of Venice Abbot Antonio Conti’s war to destroy Leibniz and rehabilitate Galileo; this is a continuation of Paolo Sarpi’s use of the “brainwashed” Galileo to guide Bacon et al. in their attacks upon Nicolaus of Cusa, Leonardo da Vinci, and Johannes Kepler. This is the issue of 1885-1886, between Cantor, on the one side, and the followers of LaPlace, Cauchy, and Moigno, on the opposing side.¹⁸ This is the mathematical, ontological, and theological issue which permeates the immediate environment of the Cantor-Franzelin exchange.

To identify the axiomatic formalities of the issue between Cantor and such followers of Galileo and LaPlace as Cauchy and Moigno, it is sufficient to focus upon the review of elementary geometry just supplied here. Look at the change in proceeding from the axiomatic basis of theorem-lattice A to that of B, or B to C, or C to D.¹⁹ From the standpoint of Aristotelian formalism, the movement from one such lattice to the higher successor is a formal-logical discontinuity, and also a mathematical discontinuity. This discontinuity, separating the axiomatic basis of one theorem-lattice from the next, is the formal reflection of an act; it is the representation of what we term in physics a true singularity. That act is the employment of the creative processes of mind, as described by Plato’s Socratic method, to discover a solution to a “One/Many” paradox of the type illustrated by the Parmenides.

This discontinuity, which has a mathematical size of virtually zero—but not zero, is a correlative of what Plato signifies by “change.” This change, this mathematical discontinuity is the root ontological referent for Cantor’s notion of the transfinite. Since Riemann’s famous Habilitation dissertation of 1854 on hypothesis, such singularities expressed as paradoxes of the formal domain of mathematics are the entry-points for the crucial issues of physics, which can be addressed efficiently only from the standpoint of physics, and not formalist mathematics as such.²⁰

In light of this kind of evidence, it is clear than the “infinite” as conceived by Aristotle and other formalists does not exist. The proof is, that every formal theorem-lattice, within whose terms such a popular misapprehension of the term “infinite” is projected by formal logic, is itself finite or, “transfinite”! Every theorem-lattice is bounded externally by a higher-order theorem-lattice, until the very conception of Plato’s Becoming reaches its upper, external boundary, defined by the Good, the location of existence of the Mosaic God of the Apostles John, Paul, et al., which latter bounds everything efficiently. Those are the mathematical, physics, and theological implications of the Cantor-Franzelin exchange, the environment within which the discussion is situated.

The fact that discovery of relatively higher-order theorem-lattices enables us to conceptualize as a single mental object the differences between the respective sets of axioms underlying two compared formal theorem-lattices, permits us to replace the commonplace, but pathological notion of an “infinite” with the notion of the boundedness, hence “transfiniteness” of that set of axioms which defines the theorem-lattice, within which latter the corresponding pathological notion of an “infinite” is situated.²¹

Cantor’s general form of solution to conceptualization of the notion of infinite in a non-pathological way, is to express the Many-ness of very large arrays within a specific theorem-lattice by a One. That One is the unified notion of the set of axioms and postulates underlying the consistency among all possible theorems of that specific theorem-lattice type.

This is the problem which Bertrand Russell, for one, attempts to circumvent by mere word-juggling, using the term “hereditary principle.” I.e., since every possible theorem of a consistent lattice is hereditarily consistent with the imputable set of axioms and postulates underlying it, that set of axioms and postulates must be construed as an “hereditary principle”; once the hereditary principle’s distinctions are understood, as distinct from that of other lattices, the notion of any infinity apparently existing within a formal lattice is expressed adequately by direct reference to the “hereditary principle.” The trouble with Russell’s version of this, and those of his followers, is that his views involve a deliberate fraud, a methodological, formalist’s fraud closely related to that of LaPlace, Cauchy, and Moigno earlier.

To understand the Cantor-Franzelin exchange adequately, one must know these background considerations. To understand Cantor himself adequately, one must return to the clean fresh air of Riemann’s 1854 paper on hypothesis.

Once one steps out of the precincts of the street mathematician, into the realm of theology, the issue between Cantor and Moigno is a replay of the continuing issue between Cardinal Nicolaus of Cusa and Aristotelian apologist John Wenck, back during the 1440’s. Not only does Cantor rightly trace his discoveries to the mathematical discoveries of Nicolaus of Cusa. That is the issue of attacks on Cusa by Pietro Pompanazzi and his students, such as Francesco Zorzi, and the later attacks upon Cusa’s method and influence by the atheists Paolo Sarpi (who deployed Galileo) and Cauchy’s mentor LaPlace.²² To pose such issues within a theological deliberation among public figures, one a cardinal, in the 1880’s, is to raise the specter of possible schism between the followers of St. Augustine (the Platonists) and the followers of Wenck and Pomponazzi (the Aristotelians). To say the least, Cantor posed a very touchy subject in his correspondence.

Georg Cantor fully in his right mind would never adopt Newton’s “hypotheses non fingo,” nor send praises of Theosophist’s hero Francis Bacon to Pope Leo XIII.

The Formalities of the Issue

Now, to conclude, identify as simply as possible the form of the issue between the followers of LaPlace and Cantor, the formalities of the Cantor-Franzelin exchange.

Cantor’s correspondence references symptomatically an issue which is as old as the beginning of modern European civilization, the issues of the principles of the founding of modern science by Nicolaus of Cusa’s De Docta Ignorantia²³ and related writings.

Once one situates observation of the act of mental-creative discovery within the formalities of classical geometry, as Cusa did in solving the ontological paradox of Archimedes’ theorems on quadrature of the circle, one has immediately two notable results. First, one has rendered the act of creative mental activity itself a subject available to conscious reflection, has rendered the creative processes of the mind intelligible. One is obliged to explore the same principle of intelligible creativity shown in such a geometry setting, to see the same quality of intelligible mental phenomenon in other areas of application.

Since the work of Paolo Sarpi’s tame gnostic, Galileo Galilei, the fraudulent tactic which the followers of Galileo’s method have employed to attempt to evade the kinds of singularities to which we have referred above, is to insist, hysterically, as Venice agent Dr. Samuel Clarke did in the Leibniz-Clarke correspondence, upon the ultimate authority of infinite series. They claim, that since infinite series may approximate all possible values within mathematical functions, mathematical discontinuities do not exist. Often, they even worship such an infinity, insisting that the unfathomable outer reaches of “infinity” are the place of residence of what Harvard Professor William James specified as the universal common root of “varieties of religious experience,” or what Sigmund Freud (or, is it “Fraud”) identified as “the oceanic feeling.”²⁴

That copying of the notion of infinite series inhering in the method of Galileo, is that same standpoint expressed by Venice’s Eighteenth-Century control agent, Abbot Antonio Conti, his accomplice Abbot Guido Grandi of Pisa, and his protégé and Grandi student Giammaria Ortes. This is the standpoint of radical empiricism, such as that of Jeremy Bentham and his followers in Britain, and also the standpoint of the French Restoration form of radical empiricism, the positivism of the followers of LaPlace and Cauchy.²⁵

Cardinal Franzelin’s abrupt termination of the correspondence with Cantor did not cause Cantor’s capitulation to British Theosophy during the late 1890’s; unfortunately, had Franzelin’s rejection of continued discussion not have occurred as it did, Cantor’s mind might not have cracked under the pressures of such London assets in Germany and France as Kronecker and his accomplices.

Cantor’s work remains a great contribution to mankind, and his efforts to clarify this issue with a representative of the Vatican are an honorable part of that. His collapse under two decades of his enemies’ aversive attempts at his behavioral modification, is an important tragedy of modern history, especially for science, but also for mankind. Cantor himself believed that his discoveries would not be properly appreciated until some time during the Twentieth Century. Generally speaking, his insight on that point was prophetic, although we must thank those, including Kurt Gödel, who kept his work alive for us today. To go forward with his contributions, it is sufficient to begin with a slight detour, to situate Cantor’s discoveries within the developments flowing through Riemann’s 1854 habilitation dissertation on hypothesis

Back to Transfinite Article

1. Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre (1897), in Georg Cantors Gesammelte Abhandlungen, ed. by Ernst Zermelo (Hildesheim: Georg Olms Verlag, 1962), pp. 282-351. The readily available English translation is that of Cambridge University-trained Philip E.B. Jourdain: Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, (1915) (New York: Dover Publications, 1955). For reason of that precedent, the Jourdain English translation of the title has been employed here. The reader is cautioned that Jourdain“s notes for the 1915 edition are rendered obsolete by Kurt Gödel’s “On formally undecidable propositions of Principia Mathematica and related systems I” (“Über formal Unentscheidbare SÜtze der Principia Mathematica und verwandter Systeme I“), in Kurt Gödel: Collected Works, Vol. I, ed. by Solomon Pfeferman et al. (New York: Oxford University Press, 1990), pp. 144-199 (including appended note by editors).

2.. On Cardinal Fanzelin’s termination of the correspondence, see Georg Cantor Briefe, ed. by Herbert Meschkowski and Winfried Nilson (Berlin: Springer-Verlag, 1991), pp. 256-257. On the subject of this correpondence and also Cantor“s depression of the 1890“s, see the same source, pps. 11-16, 252-258, 282- 285.

3.. Op. cit., p. 282.

4. . Grundlagen: über unendliche lineare Punktmannigfaltigkeiten, in Gesammelte Abhandlungen, op. cit., pp. 139-246.

5. . Op. cit., pp. 378-451 (including appended notes from Dedekind correspondence).

6. . See Meschkowski and Nilson, op. cit., passim. The Anglophile phase of Cantor’s depression erupts visibly during the approximately two-year span of time from the 1895 break in his already deeply strained intellectual relationship with Professor Felix Klein, through such 1897 events as the publication of the Beiträge and the death of Cantor’s former mentor, Karl Weierstrass. During that interval, Cantor has developed a close acquaintance with Rudolf Steiner, a member of the British Theosophist movement, a founder of the Vienna-based Theosophist periodical, Luzifer, and later founder of the German (Waldorf) spin-off of the Theosophists, the Anthroposophic movement. (The legend is that Steiner concluded that the radicalism of Bertrand Russell’s crony, the Theosophical leader and satanist Aleister Crowley, was a bit strong for customary German Kantians, and produced the altered dogma of the anthroposophs with this thought in mind.)

It was in this setting, of the association with Rudolf Steiner’s British Theosophism, that Cantor adopted the cultish view that “Theosophy-saint” Francis Bacon had actually written Shakespeare’s dramas. It must be taken into account, that all of Cantor’s creative work was grounded in the deepest rejection of everything for which Francis Bacon’s followers stand. It is clear that Cantor’s change of heart toward Bacon could have occurred only as a result of a persisting “behavior modification by aversive conditioning,” supplied by Iago-like Leopold Kronecker, et al.

Note the relationships with British agents such as Cambridge University’s Jourdain (the translator of the Beiträge), Grace Chisholm-Young, and even Cantor’s own mortal intellectual adversary, Russell himself. See also Section 4 from Professor Ernst Fraenkel’s biographical sketch, “Das Leben Georg Cantors,” in Gesammelte Abhandlungen, op. cit., pp. 469-475 on Cantor’s honors and connections in Britain from the period of his close acquaintanceship with Rudolf Steiner. The dating of Cantor’s first contact with Rudolf Steiner’s circles is not clear; what is clear is the horrifying implication of Cantor’s February 13, 1896 letter to Pope Leo XIII: “Permitte, Pontifex Maxime ... tria volumnina operum Francis Baconi addam.” The Cantor of that letter is no longer the Cantor of the Grundlagen or the earlier correspondence with Cardinal Franzelin.

7. . Impossibilité du nombre actuellement infini; la science dans ses rapports avec la foi (Paris: Gauthier-Villars, 1884). See Cantor, “Über die verschiedenen Standpünkte in Bezug auf das aktuelle Unendliche,” in Gesammelte Abhandlungen, op. cit., pp. 370-377.

8. . Ibid.

9. . Meschkowski, op. cit., pp. 252-253.

10.. Ibid.

11.. Op. cit., pp. 254-257.

12.. Op. cit., p. 258.

13.. Op. cit.

14.. Op. cit.

15.. Cantor’s repeated insistence on this during his writings of the 1880’s is indispensable for avoiding the commonplace blunders of the proverbial “usual generally recognized authorities” in their reading of both the Beiträge and these earlier writings.

16.. There is a presentation of this in numerous of this author’s writings, including Section 2 of the current “How Bertrand Russell Became An Evil Man,” Fidelio, this issue.

17.. Kurt Gödel, op. cit.

18.. LaRouche, “Evil Man,” op. cit.; Section 2, passim.

19.. Ibid.

20.. Ibid.

21.. Ibid.

23.. See Sigmund Freud, Civilization and Its Discontents, in The Freud Reader, ed. by Peter Gay (New York: W.W. Norton, 1989), p. 723ff.

24.. LaRouche “Evil Man,” op. cit.; Section 2, passim.

25.. Ibid.