**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.

Introduction

Contingence

**T**owards 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}

Cantor's Contribution to a Contingent World View

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 Religious Background

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}

The Existence of the Transfinite

Cantor's Idea of a Completed Set

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}

Transfinite Numbers in Contrast to Absolute Infinity

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 ω^{2}. 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 ( ý_{0} ), the
cardinality of the integers; the second cardinal was Aleph-One ( ý_{1} ),
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, ýý_{0}, ýý_{1},
ýý_{2}, ýý_{3}, 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.

Whether Transfinite Numbers Exist

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-Leffler^{26} Cantor wrote that
his motive in studying transfinite cardinalities was to address certain applications in
chemistry, optics, and biology.^{27}In 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 (*monad*s), 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.

** Cantor's Universe Not Necessitarian**

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.
Torrance^{48} 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**

^{1}Seminal 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*.

^{2}Thomas 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).

^{3}In 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).

^{4}An 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).

^{5}For 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.

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

^{7}Dauben, op. cit., p. 232.

^{8}Dauben 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).

^{9}Dauben, op. cit., p. 291.

^{10}From 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.

^{11}To 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.

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

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

^{14}This 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 .

^{15}City 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.

^{16}I 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.

^{17}That 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.

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

^{19}Dauben, op. cit., p. 146.

^{20}Dauben, op. cit., p. 126.

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

^{22}Quoted in Dauben, op. cit., p. 138.

^{23}Newtonian 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.

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

^{25}A 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.

^{26}quoted in Dauben, op. cit., p. 294.

^{27}Dauben, op. cit., p. 292-294.

^{28}Dauben, op. cit. p. 132.

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

^{30}See note 5.

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

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

^{33}Dauben, op. cit., p. 295.

^{34}Dauben, op. cit., p. 360.

^{35}Dauben, op. cit., p. 108.

^{36}See note 12.

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

^{38}Dauben, op. cit., p. 145.

^{39}Dauben, op. cit., p. 295.

^{40}Dauben, op. cit., p. 270.

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

^{42}Dauben, op. cit., p. 349.

^{43}See note 42.

^{44}Ecclesiastes 3: 10, 11 RSV.

^{45}See note 37.

^{46}Rudy Rucker, *Infinity and the Mind*
(Boston: Birkhauser, 1982).

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

^{48}See note 11.

^{49}Also see Rucker, op. cit., p. 117.

^{50}See note 28.

^{51}Dauben, op.cit., p. 296.