Cantor's Concept of Infinity:
Implications of Infinity for Contingence

by THE REVEREND BRUCE A. HEDMAN, Ph.D.

Department of Mathematics
University of Connecticut
West Hartford, CT 06117

From Perspectives on Science and Christian Faith 46 (March 1993): 8-16.

Georg Cantor (1845-1918) was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the impact Cantor's Christian convictions had on the development of transfinite set theory. In this paper I propose to examine how Cantor's transfinite set theory has contributed to an increasingly contingent world view in modern science. The contingence of scientific theories is not just a cautious tentativeness, but arises out of the actual state of the universe itself. The mathematical entities Cantor studied, transfinite numbers, he admitted were fraught with paradoxes. But he believed that they were grounded in a reality beyond this universe, not finally determinable by any mathematical system.

Towards the close of the twentieth century I believe that Christians are finding the climate of science to be more hospitable to our faith than did our forbearers in the nineteenth. I shall refer to Newtonian mechanics as it was developed in the eighteenth and nineteenth centuries as classical science. By modern science I shall refer to developments since and including Maxwell's electromagnetic theory and Einstein's special and general theories of relativity. Classical science sought to understand all phenomena solely in terms of particle-on-particle interactions, governed by Newton's basic laws of motion and gravitational attraction. So classical science came to regard the universe as self-contained, materialistic, and deterministic. But such a closed universe has no room for revelation, miracles, and salvation. The pursuit of scientific knowledge came to be regarded as antithetical to the Christian faith.

But this conflict was an aberration in the development of science. In this century philosophers of science have become aware that during the Enlightenment experimental science grew out of the Christian doctrine of creation.¹ More recently, the scientific revolutions which reshaped modern thinking were encouraged by Christian notions of contingence. We are indebted to the Very Reverend Professor Thomas F. Torrance for demonstrating how deeply Michael Faraday and James Clerk Maxwell were guided by their Christian belief in the universe as God's free creation, a conviction which also influenced Albert Einstein.² Modern science regards the universe as complex, subtle, and as far more open and free than did classical science.

These differences may be summarized using the word contingent. An incontingent world view regards the universe as closed, self-contained, and self-explanatory, that is, not requiring any explanation beyond itself. Such a universe would be deterministic, that is, all that occurs must necessarily have happened according to a system of fixed laws. Such a universe even taken as a whole must necessarily be the way it is, and not otherwise. As such a universe can be explained according to a system of fixed laws, it is essentially simple.

In contrast, a contingent world view regards the universe as open, as ultimately not explainable in terms of itself alone. On this view, no scientific theory can account for all phenomena. Such a universe need not necessarily be the way it is. One cannot understand phenomena through a priori reasoning alone, but must study the phenomenon itself.³ A contingent world view regards the universe as essentially complex, subtle, and mysterious. lt believes that an order may be found underlying diverse phenomena, but that this order is itself contingent, that is, always subject to further modification to embrace yet more diverse phenomena. In contrast to classical science's veneration of Newtonian mechanics, modern science regards its theories more tentatively, however beautifully they may now order known phenomena.⁴ Most scientists today readily admit the contingence of scientific theories, and increasingly more of them will admit to the contingence of the universe itself.⁵

Georg Cantor (1845-1918) was a deeply religious Lutheran whose Christian convictions consciously shaped his scholarly work, including his mathematics. His biographer Joseph Dauben wrote, "The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and the developments he gave it."⁶ Dauben has expressed his surprise that the intentional impact of Cantor's deep Christian convictions has received so little attention from historians of science.⁷

In this paper I propose to examine the following two questions. 1) How did Cantor's Christian understanding of the universe as contingent influence his development of transfinite set theory? 2) How has transfinite set theory contributed to an increasingly contingent world view in modern science? I will examine these questions under three headings, according to the influence of a contingent world view upon epistemology (third section of this paper), cosmology (fourth section), and ontology (fifth section). I am indebted to Dauben's study of Cantor for key points in the discussion.

Cantor's father, Georg Woldemar Cantor, was of Jewish descent, but was brought up in a Lutheran mission in St. Petersburg. He married Maria Anna B–hm, an evangelical convert from Roman Catholicism. Their six children, of whom Georg was the eldest, were baptized there in B–hm in the Evangelical Lutheran Church in St. Petersburg.⁸ Georg Woldemar was very devout, and gave disciplined religious instruction to all his children. Georg Cantor was confirmed in the Lutheran church at age fifteen.

Throughout the rest of his life Georg Cantor firmly held to the Christian faith his father had instilled in him. During his university studies Cantor felt a deep calling from God to study philosophy and mathematics, rather than more lucrative pursuits. His faith sustained him during long years of rejection when the mathematical establishment dismissed his concept of the transfinite. When weaker men would have abandoned their work, Cantor persevered, not only because he believed that the transfinites had been revealed to him, but moreover because he felt a calling to spread the truth about God's creation for the benefit of both the Church and the world.⁹ He wrote in a letter to a Dominican priest in early February 1896, From me, Christian philosophy will be offered for the first time the true theory of the infinite.¹⁰

Before dealing with the impact of transfinite numbers on the modern scientific world view, I need to discuss their existence. Traditionally Christian theology attributed the characteristic infinite to God alone.¹¹ Thomas Aquinas gave this classical formulation. Things other than God can be relatively infinite, but not absolutely infinite.¹² A quantity is relatively or potentially infinite, if it is simply unbounded. The absolutely or actually infinite contains within itself already an infinite magnitude. Previously, mathematicians had spoken of infinity only as potential, not actual; only as unbounded. In 1831 Gauss wrote, The infinite is only a facon de parler in which one properly speaks of limits.¹³

Cantor introduced into mathematics the notion of a completed set, so that the integers, for example, could be considered together as a set in themselves, and so as a completed infinite magnitude. Only by conceiving of the integers as a whole entity, (as a Ding f¸r sich) could Cantor define the first transfinite number, which he denoted by a lower case omega (ω), in contradistinction to the familiar sideways eight infinity symbol (.), which had only meant unbounded. Cantor saw a precedent for this intellectual step in Plato's treatment of the Aone/many problem.¹⁴ More influential was Augustine's argument, often quoted by Cantor, that all infinity is in some ineffable way made finite to God, for it is comprehended by his knowledge.¹⁵

The integers considered as a completed set were Cantor's first transfinite number, ω. He went on to construct further transfinite numbers recursively.¹⁶ Thus, the next transfinite number was ω + 1 = ω U (ω) , and next, ω + 2 = ω + 1 U (ω) + 1}, etc. Then the sequence ω, ω + 1, ω + 2,... was taken as a completed set, denoted by ω2. Similarly, the sequence ω2, ω3,ω4...was completed as ωω, or ω². Similarly the sequence ω^2,ω^3,ω^4... was completed as ω^ω. Thus, Cantor built up an ever-broadening hierarchy of what he called ordinal numbers.

Cantor's pivotal discovery, which lends ordinal numbers their interest, was that some ordinals, though infinite in size, are nevertheless smaller than other ordinals. Cantor said two ordinal numbers were of the same cardinality (that is, size, or power) if they could be put into one-to-one correspondence with each other. Cantor's epochal discovery was that the natural and the real numbers were of different cardinality. More generally, call the set of all subsets of a set S - the power set P(S). By his now-famous diagonalization argument,¹⁷ Cantor showed that P(S) was a higher cardinality than S, that is, P(S) was too numerous to be put into one-to-one correspondence with S. Cantor distinguished ordinals of different cardinality as cardinal numbers, and denoted these by subscripting the Hebrew letter Aleph. Thus, the first cardinal was Aleph-Null ( ý₀ ), the cardinality of the integers; the second cardinal was Aleph-One ( ý₁ ), at most the cardinality of the real numbers. Cantor's discovery that the cardinality of P(S) is greater than the cardinality of S guarantees an unending hierarchy of cardinal numbers, ýý₀, ýý₁, ýý₂, ýý₃, ýý_n, Y, ýý_n+1 Y. Cantor denoted the class of all cardinals by the Hebrew letter Taw.

Note that Cantor built up transfinite numbers from below, by constructing a larger ordinal out of a smaller. From the beginning of his work he realized that infinity cannot be approached from above. Cantor was keenly aware of the paradoxes inherent in such constructions as the set of everything, the set of all sets, and even Taw. Such totality Cantor called Absolute Infinity; it is beyond all mathematical determination, and can be comprehended only in the mind of God. Cantor's distinction between transfinite numbers and Absolute Infinity had a profound impact on our modern contingent world view, which I will examine under Ontology below.

Cantor distinguished three levels of existences: 1) in the mind of God (the Intellectus Divinum); 2) in the mind of man (in abstracto); and, 3) in the physical universe (in concreto.) Cantor believed that Absolute Infinity exists only in the mind of God. But he argued that God instilled the concept of number, both finite and transfinite, into the mind of man. Cantor frequently appealed to their existence as eternal ideas in the mind of God as the basis for the existence of the transfinites in the mind of man.¹⁸ I will pursue the implications of this appeal for our understanding of contingent rationality under Epistemology below. Cantor adamantly defended the existence of the transfinites in abstracto, even arguing that God had put them into man's mind to reflect his own perfection.¹⁹ Cantor advanced infinite series representations of irrationals to claim that their existence was equivalent to that of the transfinites.²⁰

Cantor was a realist concerning the relationship between the ideal and physical reality of numbers. The existence of numbers in abstracto he called their intra-subjective reality, and their existence in concreto their trans-subjective reality. There was a created correspondence between these two realities which I will discuss under Epistemology below. Although he denied that transfinite numbers have a trans-subjective reality in the duration or extent of the physical universe, Cantor did follow Leibnitz in believing that there were a transfinite number of elementary particles (monads) in the physical universe. I will discuss this further under Cosmology below.

Epistemology

Contingent Rationality

Newtonian physics enjoyed tremendous prestige in the eighteenth and nineteenth centuries, as it united under one system both terrestrial and celestial mechanics. The Newtonian world view sought to explain everything in the universe in terms of particle-on-particle interactions governed according to these physical laws. But such a philosophy must struggle to account for the role of the mind in such a mechanistic, deterministic universe. Why should mental, mathematical abstractions correlate with the physical world? This question spurred the developments advanced by Locke, Berkeley, and Hume in the eighteenth century.

The Newtonian system was finally closed in upon itself by Immanuel Kant, who argued that the mind does not experience the physical world itself, the noumena, but only its own sense perceptions, the phenomena. Scientific laws are not statements about physical reality itself, but are only the mind's own ordering of its sense perceptions. The mind has innate patterns, the a priori, according to which it orders its perceptions. Kant considered Euclidean geometry, for example, as a priori knowledge. Kant lent classical science its final, self-contained, incontingent character by closing it off from not only metaphysics but from physical reality itself.

This Kantian interpretation floundered on the scientific revolutions of the late nineteenth and early twentieth centuries.²¹ If science is no more than the mind's projection onto its sense, how could such thorough-going scientific revolutions arise? The first tremors shaking this Kantian foundation came from mathematics, with the discovery of non-Euclidean geometry. In a letter dated 1885 the Swedish mathematician G–sta Mittag-Leffler wrote to Cantor that his transfinites were as revolutionary as non-Euclidean geometry.²² Maxwell's electromagnetic field introduced a fundamentally non-Newtonian interaction.²³ Einstein's theory of relativity revolutionized Newton's notions of space as a container and of time as absolute, and hence Kant's interpretation of space and time as forms of perception.

Thus, modern science has moved away from a positivist toward a realist interpretation of the reciprocity between the mind and the physical universe. I believe that modern recognition of contingent rationality can be described in two movements. First, reason cannot understand nature a priori, but must go to nature itself and ask questions that nature may disclose itself. Indeed, the ancient Christian doctrine of contingence is the philosophical basis of experimental science.²⁴ Out of itself the physical universe suggests patterns to the mind. Second, there is a created harmony between these mental patterns and the physical universe so that later mathematical deductions correlate with further physical phenomena. Modern science simply accepts this correlation without pursuing an explanation.²⁵ Cantor exemplified both of these movements, and so furthered modern science's grasp on the contingent nature of rationality.

Cantor Grappled With Physical Reality

Kant argued that the mind was not informed by the physical world, but only imposed its own patterns upon its perceptions. Cantor was explicitly opposed to any Kantian interpretation of science, and maintained passionately that the transfinites were not his mind's own invention, but were suggested to him through physical considerations. In 1872 Cantor first demonstrated the uniqueness of the trigonometric series representation of a function. This, then, he generalized over intervals with increasingly infinite points of discontinuity. This led him to consider the nature of the continuum and of continuous motion. In a letter to Mittag-Leffler²⁶ Cantor wrote that his motive in studying transfinite cardinalities was to address certain applications in chemistry, optics, and biology.²⁷In studying continuity Cantor made his epochal discovery of the nondenumerability of the real numbers, from which modern set theory has sprung. The counter-intuitive, even paradoxical, properties of the transfinites argue against a Kantian a priori, and suggest that physical reality has impinged upon the mind from outside itself.

I cite Cantor's interpretation of his own work as an example of what I call incarnational mathematics. There is a created rationality embedded in our minds and within the physical universe. The mathematician possesses not merely a mind, but a mind embedded, incarnated, if you will, into the physical world through his body. I believe history shows that the most productive mathematics have been suggested by physical considerations. After a period of abstract development, mathematics has often been refocused by physical applications of its abstractions.

The Applicability of Mathematics

As mentioned above (page 11), Cantor distinguished between the intra-subjective and the trans-subjective existence of all numbers, finite and transfinite. But Cantor believed these dual realities were always found together. This correspondence between the physical and ideal aspects of numbers Cantor believed came from a unity in the universe itself.²⁸ Cantor believed in a created harmony between the mind and the physical world, so that mathematical deductions from patterns first suggested by the physical world should reapply to further phenomena. He thought that the transfinites would shed light on the ultimate constitution of matter, which would benefit physics, chemistry, and biology. History has vindicated his expectations, though along different paths. Cantor's set theory laid the foundation for analysis. His discovery that n-dimensional space is of the same size for all n spurred the study of topology, which has given us fiber optics, to name one benefit.

But this created harmony is contingent. In contrast to Hellenistic philosophy, the human mind is not a divine spark, or an actual piece of God's own mind. This rationality in our minds and in the physical world is not absolute and self-dependent, but is only a created reflection of an Ultimate Rationality upon which it depends. So no scientific deduction must necessarily be so, nor is any scientific theory beyond revision. Cantor produced a classic example of contingent rationality when he drew the distinction between transfinite numbers, which exist in the human mind, and Absolute Infinity, which is beyond all human determination, and exists only in the mind of God.

Cosmology

Creation

Classical science regarded the physical universe as incontingent, as self-contained and self-explanatory. An incontingent universe must, therefore, be necessarily infinite in duration and extent. Otherwise, the questions as to what came before and what lies beyond have no answer within that incontingent universe itself. The eternity of the universe has been called the first article of the secular faith.²⁹

Cosmology is the bellwether of major paradigm shifts in a culture.³⁰ Modern science has returned to a more contingent world view of the universe as finite in duration and extent. Steve Hawking has called Big Bang cosmology one of the great intellectual revolutions.³¹ Furthermore many modern cosmologists seriously doubt that only one type of universe was logically possible, but rather argue that there were arbitrary elements in the composition of both the structure of the universe and its fundamental constants.³² The ancient Christian doctrine of creation ex nihlio regards the physical universe as God's free creation whose structure is determined by no necessity nor constraint, but is contingent only upon God's will. Cantor's thinking was shaped by his Christian belief in the universe as created.

Cantor's Universe Finite in Duration and Extent

Cantor was explicitly opposed to the prevailing materialism of his scientific community, which regarded the physical universe as eternal and unbounded. From his earliest papers Cantor stressed that transfinite numbers were of no aid to the materialist, positivist, or pantheist.³³ In a letter to K.F. Heman dated 1887 Cantor promised to show that in fact transfinite numbers could demonstrate the impossibility of eternal time, space, and matter, though such arguments, if ever written out, have not survived.³⁴ Cantor was proud to be the only indeterminist on faculty at the University of Halle. Interestingly, Cantor challenged the existence of objective or absolute time in advance of relativity theory.³⁵

For many years the only encouragement Cantor received for his work on the transfinites was from Roman Catholic scholars. Their support was at first tentative, until Cantor made certain basic distinctions. Christian theology had taught that infinity was an attribute of God's alone.³⁶ Hellenistic cosmology identified God as the soul of the world and the world as the body of God. These pantheistic notions were reintroduced by Spinoza, whose monistic philosophy of substance conceived of God as the infinite self-generating substance (natura naturans) from which the world (natura naturata) is derived. Any concrete, temporal infinity was presumably identified with God's infinity, and so suspected of pantheism.

The first theological paper to appeal to Cantor's transfinites was written in 1886 by a neo-Thomist, Fr. Constantin Gutberlet.³⁷ Gutberlet was concerned to show that actual, completed, mathematical infinity did not challenge the unique, absolute infinity of God's existence. Yet he disagreed with Cantor as to admitting the actual infinite into the created order. Whereas Cantor denied the infinity of the universe in duration and extent, he did believe, following Leibnitz, that there were an infinite number of elementary particles (monads), and so that the transfinites were realized in concreto. In a letter dated 1886 to Cardinal Johannes Franzelin, Gutberlet's teacher, Cantor made the distinction between Absolute Infinity, as eternal and uncreated, reserved for God and his attributes, and the Transfinitum (the transfinite numbers), as created in abstracto and in concreto.³⁸ Franzelin approved of this distinction as removing any threat to orthodoxy. As Franzelin was a leading Jesuit philosopher and papal theologian to the Vatican Council, Cantor took his approval as an imprimatur for his work. Cantor further argued that the real existence of transfinites in the created order reflects the perfection of the infinite nature of God's being.

An incontingent world view regards the universe as having a necessary structure, as being uniquely determined by just the requirement of self-consistency. All phenomena in principle could be deduced from its system of basic laws. A contingent universe does not contain within itself a sufficient explanation of itself, and so cannot be understood simply by a priori reasoning. In his writings about the nature of the universe Cantor was deeply conscious of its contingent character. Following Leibnitz, Cantor thought of the universe as being built up from two kinds of elementary units: corporeal (matter) and ethereal (ether) monads. Cantor held that transfinites exist in concreto in a nevertheless temporal, bounded universe, because there are a transfinite number of such monads. Cantor further believed that the cardinality of the corporeal monads was Aleph-Null, and of ethereal monads was Aleph-One, his "First World Hypothesus." But in spite of his philosophical investment, Cantor was careful to stress that God did not necessarily have to create the universe in this or any other way.³⁹   The existence of the transfinities in the mind did not even necessarily depend upon their realization in the physical universe..

Ontology

Contingent Order

Classical science regarded the universe as self-contained, and hence as self-explanatory. It had to be understood out of itself, and had to contain within itself a sufficient explanation of itself. Many believed that all phenomena could ultimately be explained by the Newtonian laws, which were highly esteemed as expressions of the fundamental structure of physical reality. The revolutions of modern science, particularly the electromagnetic field and relativity theories, showed the naivete of this interpretation. Newtonian mechanics was not simply falsified, but came to be seen as a limited case of a far wider understanding. Modern science regards the universe as far more complex than ever imagined. Albert Einstein said, God does not wear his heart on his sleeve.

Therefore, modern science regards its theories more provisionally than did classical science. No matter how true to known data, any scientific theory is considered as tentative, as a limited case of a wider reality yet to be discovered. The enterprise of modern science may be thought of as a sequence of concentric circles, like ripples radiating out from a pebble thrown into a pond, embracing a yet larger understanding of reality, but never all of it.

I believe that this interpretation of scientific theories illustrates an ontological conclusion as to the nature of reality itself. All theories are provisional, because the universe cannot be understood out of itself, and depends upon an explanation beyond itself. In other words, the contingence of scientific theories arises from the contingence of the universe itself. Since the existence of the universe depends upon a reality beyond it, no scientific theory, which is of course couched only in terms taken from within the universe itself, can finally explain everything in that universe. Thus, the order we find in the universe in contingent. I want to argue that not only did Cantor exemplify this attitude, but that his transfinites have consequently stimulated this understanding of the contingent order within the universe.

Contingence of Scientific Theories

Cantor's transfinite set theory changed the way mathematics thinks about itself. Joseph Dauben wrote, Cantor's infinite had shaken the traditional faith in mathematics' everlasting certitude.⁴⁰ Cantor believed that he was studying mathematical entities which existed apart from and beyond any mathematical system. He thus had no qualms about embracing the paradoxes which arise in transfinite set theory. Although Cantor did not explicitly write about these paradoxes until 1895, he seems to have been aware of them in his first book devoted solely to set theory, published in 1883. There he defined a set as a collection which could be taken as a completed whole (Ding f¸r sich). This requirement of completion precluded self-generating sets like the set of all ordinals (Omega), the set of all cardinals (Taw), or the set of sets. Cantor accepted the impossibility of analyzing mathematically the entire succession of the transfinite.

Yet Cantor was absolutely convinced about the real existence of the transfinites. I believe that Cantor would not have sympathized with the formalists of the next generation of mathematicians, who sought to reduce all mathematics to a logical system. He believed mathematics was simply too large for that. In contrast to these formalists, the spirit of Cantor's approach was vindicated by G–del's work on incompleteness in 1931.

Cantor would not have been surprised, I think, to learn that any system large enough to include the integers would contain propositions whose truth was undecidable within that system. From a realistic point of view, G–del vindicated his conviction that mathematics itself is far richer than any formal systemization of it. As mathematics is the model for other scientific disciplines, I believe that this harbingers the incompleteness, and hence contingence, of any scientific theory.

One specific case is worthy of note. Cantor spent years unsuccessfully trying to prove his Continuum Hypothesis, that the cardinality of the geometric continuum is the smallest uncountable cardinal number, that is, P(Aleph-Null) = Aleph-One. In 1936 G–del showed that Cantor s Continuum Hypothesis was at least consistent with set theory, and in 1963 Paul Cohen showed it was in fact independent. Thus, Cantor's prize conjecture turned out to be far richer than he had ever guessed. Cohen has suggested that the continuum may be larger than all Alephs.⁴¹ In analogy with the Fifth Postulate, I suggest that someday different physical situations may be found, one of which will be described by Cantor's Continuum Hypothesis, and the other by a richer continuum.

Contingence of the Universe Itself

The contingence of scientific theories is not just a cautious tentativeness, but arises out of the actual state of the universe itself. The mathematical entities Cantor studied, which existed apart from any mathematical system, were themselves grounded in a reality beyond this universe. The transfinites were fraught with paradoxes which Cantor believed the finite mind could never understand. But he relied on the Divine Intellect as the certain repository of the Transfinitum.⁴² This Absolute Infinity was the ground for the transfinites, but was itself mathematically indeterminable. Were it determinable, it would have then been limited.

The character of infinity forced the contingence of the universe onto Cantor's thinking. Joseph Dauben considered the influences of Cantor's religious views upon his creative process in discovering the transfinites. He wrote, "One is tempted to wonder if this view of God's role in ensuring the reality and existence of Cantor's Tranfinitum was responsible for his discovery of the contradictory nature of that very concept."⁴³ I want to argue further that Cantor's discovery of the paradoxical character of the transfinites points to the contingence of the universe itself. That the transfinites cannot be completely understood out of themselves is a specific example illustrating that the universe cannot be understood out of itself. Cantor thought of the infinite ascent of ever-increasing transfinite numbers as an appropriate symbol for the absolute. Likewise, I think such a picture is a fit analogy for the scientific enterprise in a contingent universe. In the words of Ecclesiastes, I have seen the business that God has given the sons of men to be busy with. He has made everything beautiful in its time; also he has put eternity into man's mind, yet so that he cannot find out what God has done from the beginning to the end.⁴⁴

Open Questions

Anselm's Ontological Argument

Anselm argued that God is that of which nothing greater can be conceived. Can this be reworded more pointedly using Cantor's transfinites? Gutberlet used a similar argument. But in the Absolute Mind the entire sequence is always in actual consciousness, without any possibility of increase.⁴⁵ Again, this is parallel to the Reflection Principle, which says that the Absolute should be totally inconceivable. Otherwise, if the Absolute is the only thing having a certain conceivable property, then the Absolute can be conceived as the only thing with this property.⁴⁶

Applying G–del's Theorem

Stanley Jaki appears as the first to have developed G–delian implications for cosmology.⁴⁷ Yet, G–del's Incompleteness applies only to systems large enough to contain the integers. So to apply G–del's Theorem in this context one must assume the integers are included in the universe at least in abstracto. This argument should be developed for finitists like Prof. Torrance⁴⁸ who nevertheless wish to appeal to G–delian results.

Zeno's Paradoxes

Cantor was motivated by the nature of the continuum and continuous motion. In short, continuous motion is possible because there are more real numbers than rational, that is P(Aleph-Null) =  Aleph-Null. This should relate to Zeno's paradox of The Arrow in Flight.⁴⁹

Disproving the Eternity of Space, Time, and Matter

Cantor claimed that he could use transfinite numbers to argue against the eternity of space, time, and matter, but apparently he never wrote the arguments down.⁵⁰ It would be interesting to try to reconstruct them, along the lines of his arguments against infinitesimals.⁵¹

Artificial Intelligence

Devotees of artificial intelligence model human thinking after Turing machines. Yet such can involve even potentially only a countable number of steps. Can one argue that from the uncountability of the real numbers that the human mind can fathom a reality larger than Turing machines can accommodate?

Kantian Philosophy

As suggested by a referee, another interesting direction would examine the effect Canton's mathematics of infinity, along with non-Euclidean geometry, had on the interpretation of Kantian philosophy in the twentieth century.

©1993

 NOTES

¹Seminal articles were written by M. B. Foster in Mind, xliii (1934), pp. 446 ff.; xliv (1935), pp. 439 ff; xlv (1936), pp. 1 ff; also, M. B. Foster, Mystery and Philosophy, (1957), pp. 87 ff; John Baillie, Natural Science and the Spiritual Life (1950), pp. 20 ff; W. A. Whitehouse, Christian Faith and the Scientific Attitude, (1951), p. 60 f. Also A. N. Whitehead, Science and the Modern World and Ian Barbour, Issues in Science and Religion.

²Thomas F. Torrance, Christian Theology and Scientific Culture (New York: Oxford University Press, 1981); also Transformation and Convergence in the Frame of Knowledge, (Grand Rapids, Eerdmans, 1984).

³In the history of science the Christian doctrine of the contingence of the universe, as a free creation by God, separate from yet dependent upon him, spawned experimental science in a way that speculative Greek philosophy never could. For an excellent account of the doctrine of contingence, see Thomas F. Torrance, Divine and Contingent Order, (New York: Oxford University Press, 1981).

⁴An example of modern physics' more humble approach to its own understanding of the universe is Einstein's reply to Helmholtz. Concerning the latter's conviction that the universe could be explained entirely in terms of one theory, Newtonian mechanics, Einstein replied, "The view appears dull and naive to a twentieth-century physicist. It would frighten him to think that the great adventure of research could so soon be finished, and an unexciting if infallible picture of the universe established for all time." Albert Einstein and Leopold Infeld, The Evolution of Physics, (New York: Simon Schuster, 1938), p. 58.

We have Thomas Kuhn to thank for making us aware of the progressive nature of scientific revolutions. T. S. Kuhn, The Structure of Scientific Revolutions, (Chicago, 1962).

⁵For a discussion of contingent world views among modern cosmologists, see Bruce Hedman, "Mathematics, Cosmology, and the Contingent Universe," Perspectives on Science and Christian Faith, vol. 41, no. 2 (June 1989), p. 99-103.

⁶Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, (Cambridge, MA: Harvard University Press, 1979), p. 291.

⁷Dauben, op. cit., p. 232.

⁸Dauben corrects numerous errors in E.T. Bells' popular biography of Cantor, the chief being that Cantor was not Jewish but Lutheran. Dauben also treats more sympathetically the nervous breakdowns Cantor suffered later in life. He apparently suffered from manic depression, and his first brief bout occurred when he was 37, and his second at 53. He was frequently in Halle's Nevenklinik towards the end of his life. Dauben points out that Cantor died in hospital only because his family was unable to take him home due to the exigencies of the First World War. cf. E. T. Bell, Men of Mathematics, New York: Simon & Schuster, 1937).

⁹Dauben, op. cit., p. 291.

¹⁰From a letter dated February 15, 1896, from Cantor to Esser. In Herbert Meschkowski, Aus den Briefb¸chern Georg Cantors," Archive for History of Exact Sciences, 2 (1965), p. 503-519.

¹¹To offer an autobiographical footnote, I became interested in the Christian interpretation of the transfinite after a conversation with Prof. Torrance in April 1987, when he expressed his view that infinity simply cannot exist in a contingent universe. As a mathematician, I was convinced nevertheless that the concept of infinity played a key role in shaping the world view of modern science.

¹²Thomas Aquinas, Summa Theologicae, (ed. Anton Pegis), (New York: Random House, 1945), Ia, Q.7, a.2.

¹³K.F. Gauss, Briefwechsel zwischen C. F. Gauss und H.C. Schumacher (C.A.F. Peters, ed.) (Altona, G. Esch) vol II, p. 269.

¹⁴This wholeness Plato called mikton in the Philebus. "All things that are even said to be consist of a one and a many, and have in their nature a conjunction of limit and unlimitedness." (16d) The Collected Works of Plato (eds. E. Hamilton, H. Cairns) (Princeton, Princeton University Press, 1961), p. 1092 .

¹⁵City of God Book 12, chapter 18, quoted from The Nicene and Post-Nicene Fathers (ed. Philip Schaff) (Grand Rapids: Eerdmans, 1979) vol. 2, p. 238.

¹⁶I can only sketch here the briefest description of Cantor's work. For more detail accessible to the general reader I recommend the following two essays:

"Infinity," by Hans Hahn, in The World of Mathematics, edited by James R. Newman (New York: Simon Schuster, and 1956), vol. 3, p. 1593-1611.

"Beyond the Finite", Great Moments in Mathematics (after 1650), Howard Eves (MA: 1981), p. 159-170.

¹⁷That the power set of a set even exists requires a special axiom, as set theory has come to be formulated in modern times. See Paul Halmos, Naive Set Theory, (New York: Van Nostrand Reinhold, 1960) p. 19.

The relation between the natural and real numbers poses a thorny problem in modern mathematics. The real numbers at least contain the power set of the naturals, but how much richer they may be is a question of the continuum Hypothesis. For a more detailed discussion of power sets and the Continuum Hypothesis, see Halmos, op. cit, p. 92, 102.

¹⁸Dauben, op. cit., "Cantor's Correspondence with Hermite Concerning the Nature and Meaning of the Transfinite Numbers," p. 228-232.

¹⁹Dauben, op. cit., p. 146.

²⁰Dauben, op. cit., p. 126.

²¹For an excellent discussion see "Christianity in Scientific Change", in Thomas F. Torrance, Christian Theology and Scientific Culture, op. cit., p. 11-39.

²²Quoted in Dauben, op. cit., p. 138.

²³Newtonian mechanics understood force as action along a line though the centers of interacting bodies. Maxwell's fields acted perpendicularly to this direction. Despite early prodigious efforts, this could not be explained satisfactorily in terms of Newtonian dynamics. The history of this attempt is well chronicled by William Berkson in Fields of Force: The Development of a World View from Faraday to Einstein, (London: 1974). An impeccable witness to the revolutionary non-Newtonian character of the Maxwellian field is Einstein himself, "The Reality of the Field," in A. Einstein and L. Infeld, op. cit., p. 148-156.

²⁴For an excellent discussion of this, see "Theological and Scientific Worldviews", in Thomas F. Torrance, Divine and Contingent Order, op. cit., p. 62-84.

²⁵A classic statement of this inexplicable correlation between mathematics and the physical sciences is Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics, vol iii (190), 1-14.

²⁶quoted in Dauben, op. cit., p. 294.

²⁷Dauben, op. cit., p. 292-294.

²⁸Dauben, op. cit. p. 132.

²⁹Jaki, Stanley L., Cosmos and Creator, (Edinburgh Scottish Academic Press), p. 108.

³⁰See note 5.

³¹Hawking, Stephen W., A Brief History of Time (New York: Bantam, 1988) p. 39.

³²Barrow, John D., The World Within the World, (London: Oxford University Press, 1988), p. 323.

³³Dauben, op. cit., p. 295.

³⁴Dauben, op. cit., p. 360.

³⁵Dauben, op. cit., p. 108.

³⁶See note 12.

³⁷Gutberlet, Constantin, Das Problem des Unendlichen, Zeitschrift f¸r Philosophie und philosophische Kritik, 88 (1886), p. 179-223.

³⁸Dauben, op. cit., p. 145.

³⁹Dauben, op. cit., p. 295.

⁴⁰Dauben, op. cit., p. 270.

⁴¹Paul Cohen, Set Theory and the Continuum Hypothesis, (New York: W.A. Benjamin, 1966), p. 151.

⁴²Dauben, op. cit., p. 349.

⁴³See note 42.

⁴⁴Ecclesiastes 3: 10, 11 RSV.

⁴⁵See note 37.

⁴⁶Rudy Rucker, Infinity and the Mind (Boston: Birkhauser, 1982).

⁴⁷Stanley L. Jaki, The Relevance of Physics, (Chicago: University of Chicago Press, 1966), p. 127-130.

⁴⁸See note 11.

⁴⁹Also see Rucker, op. cit., p. 117.

⁵⁰See note 28.

⁵¹Dauben, op.cit., p. 296.