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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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A true departure from Greek precedents was the
manner in which mathematics of the nineteenth cen-
tury set out, in earnest, to deal with infinity—especially
with the infinitely large—by confrontation and actu-
alization. One such development, which we will briefly


sketch in the next section, was inaugurated in geome-
try, that is in the theory of space structure; so-called
“open” spaces were boldly “closed off” by addition
of ideally conceived “infinitely distant” points that
were operationally created for such purposes. Inter-
nally these were important events which affected the
course of mathematics profoundly, even if philosophers
did not become aware of them; but externally the
dominant and spectacular development was Georg
Cantor's creation of the theory of sets and of transfinite
numbers. It had a wide appeal, and an enduring effect,
outside of professional mathematics too. Cantor's work
was not only a creation, it was a movement. As of a
sudden, infinity ceased to be an object of frequently
aimless and barren ideational speculations, and it be-
came a datum of refreshingly efficient operational
manipulations and syllogizations. The movement
brought to the fore novel thought patterns in and out
of mathematics, and it helped to create the tautness
of syntax in and out of analytical philosophy. Also our
present-day “New Mathematics,” which—at any rate
in the United States—is being introduced on all levels
of pre-college schooling, is a delayed response to a
permanent challenge which has been emanating from
Cantor's theory from the first.

But before these Victorian achievements, that is, in
the overlong stretch of time from the early Church
Fathers to the early nineteenth century, and even
during the ages of the scientific revolution and of the
Enlightenment, mathematical developments regarding
infinity were, on the whole, excruciatingly slow.
Newton, Leibniz, Euler, Lagrange, or even Carl Fried-
rich Gauss, would not have been able to express satis-
factorily, in words of theirs, when an infinite series is
convergent and when not. As we have already stated,
John Wallis introduced in 1656 our present-day symbol
“∞” for infinitely large, and he began to operate with
it as if it were one more mathematical symbol. This
can be done, to an extent. But, from our retrospect,
for about 150 years the operations with the symbol
were amateurishly and scandalously unrigorous. How-
ever, long before that, in the great mathematical works
of Euclid, Archimedes, and Apollonius, of the third
century B.C., there were well-conceived convergence
processes, which, within their own settings, were han-
dled competently and maturely. It must be quickly
added however, that this mature Greek mathematics
did not have the internal strength to survive, but was
lost from sight in the obscurity of a general decline
of Hellenism, whereas the mathematics of the seven-
teenth and eighteenth centuries, however beset with
shortcomings of rigor, has been marching from strength
to greater strength without a break.

It had been a tenet of Aristotle that there cannot
be anything that is infinite in actuality, meaning “that
no form of infinite exists, as a given simultaneously
existing whole” (Ross, Aristotle, p. 87). But 22 centuries
later, Georg Cantor retorted, boastfully, that his find-
ings clearly controverted the tenet. Cantor also ad-
duced illustrious predecessors of his, notably Saint
Augustine, who had anticipated the actual infinity of
his, even as it applies to natural numbers (Cantor,
Gesammelte..., pp. 401-04 and other passages).
These statements of Cantor are misleading, and we will
briefly state in what way.

On the face of it, Cantor was right in affirming that
there is an anticipation of the first transfinite cardinal
number in Saint Augustine's De civitate Dei, especially
in the chapter entitled “Against those who assert that
things that are infinite cannot be comprehended by the
knowledge of God” (Book 12, Ch. 18). However, this
anticipation and the others which Cantor adduces,
were ideations only, and were made and remained at
a considerable distance from mathematics proper. But
Cantor's theory of sets was produced in a spirit of truly
“abstract” mathematics; it quickly moved into the
central area of operational mathematics and has re-
mained there ever since. Within theological and philo-
sophical contexts, actual infinity, however exalted, is
hierarchically subordinate to a supreme absolute of
which it is an attribute. But in set theory, infinity,
although a property of an aggregate, is nevertheless
mathematically autonomous and hierarchically su-
preme; like all primary mathematical data it is self-
created and self-creating within the realm of mathe-
matical imagery and modality.

In some of his writings Cantor reflects on the nature,
mission, and intellectual foundation of his theories, and
these reflections create the impression that Cantor's
prime intellectual motivation was an urge to examine
searchingly Aristotle's contention that infinity can exist
at best only potentially, and never actually. But Can-
tor's mathematical work itself, if one omits his self-
reflections, suggests a different kind of motivation, a
much more prosaic one. It suggests that Cantor's theory
evolved out of his preoccupation with an everyday
problem of working mathematics, namely with Rie-
mann's uniqueness problem for trigonometric series.
Some of Riemann's work, for instance his momentous
study of space structure, is clearly allied to philosophy.
But the problem of technical mathematics which at-
tracted Cantor's attention was not at all of this kind.
There was nothing in it to stimulate an Ernst Cassirer,
Bertrand Russell, A. N. Whitehead, or even Charles
S. Peirce or Gottlob Frege. Also, the nature of the
mathematical problem was such, that Cantor was led
Sto conceive ordinal numbers first, cardinal numbers
next, and general aggregates last (Cantor, Gesammelte
..., p. 102, editor's note 2). But in a later systematic
recapitulation (ibid., pp. 282-356), which is “philo-


sophically” arranged, the order of concepts is reversed.

We have dwelt on this, because, in our view, the
actual infinity as conceived by Cantor, is entirely
different from the actual infinity as conceived by Aris-
totle, so that there is no conflict between Aristotle's
denial and Cantor's affirmation of its existence. In
support of this view we observe as follows: according
to Cantor (Gesammelte..., pp. 174-75), Aristotle had
to deny the existence of an actual infinity, simply
because Aristotle was not intellectually equipped to
countenance the fact that if n is a finite number and
α a transfinite number, then α “annihilates” n, in the
sense that

n + α = α.

Cantor observes that, contrary to what Aristotle may
have thought, this is a true and important fact, and
he derides Aristotle for not grasping it but finding
something incongruous in it. Cantor elaborates on this
fact by further noting that if α is a number of ordinal
type, and if the order of the addends n and α is in-
verted, then n is not annihilated, because, in fact
α > n = α

Also, Cantor interprets all this to imply—in all serious-
ness—that if a finite number has the temerity of placing
itself in front of an infinite ordinal number α then it
suffers annihilation, but if it has the prudence of rang-
ing itself in the rear of an infinite ordinal α then its
existence is mercifully spared.

This bizarre interpretation, however alluring for its
boldness, must not be allowed to detract from the fact
that Aristotle himself, in the given context (Physica,
Book 3, Ch. 5; 204b 12-20), to which Cantor refers
(he actually refers not to this passage in the Physica,
but to a less “authoritative” near-duplication of it in
Metaphysica, Book II, Ch. 10), speaks not of number
(arithmos), or even magnitude (megethos), but of
“body” (soma, σω̃μα ), which he expressly specifies to
be an elementary constituent of matter, like fire or air.
Aristotle asserts that such a body cannot be infinite,
because if it were, then the addition (or subtraction)
of a finite amount would not affect the sum total. This
assertion, whatever its merit, is a statement about
physics or natural philosophy, and not, as Cantor mis-
leadingly presents it, a statement about technical
mathematics. One can easily formulate a statement
which would sound very similar to the assertion of
Aristotle, and which a present-day physicist might
accept, or, at any rate, not find unreasonable. Thus,
a present-day physicist might reason that it is incon-
gruous to assume that the total energy of the universe
is infinite. In fact, if it were infinite, the addition or
subtraction of a finite amount of energy would not
change the total amount of energy, and the law of the
conservation of energy—if our physicist generally sub-
scribes to it—would become pointless when applied
to such a universe as a whole. It is true that nowadays
the law of conservation of energy, although adhered
to in laboratory physics, is not always observed in
cosmology. Thus in present-day cosmological models
with “continual creation of matter” the total energy
is nonfinite and the law of the conservation of energy
is not enforced. But the infinity involved in these
models leans more towards Aristotle's potentiality than
Cantor's actuality, and is certainly not as fully “actual”
as in Cantor.