Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*V. THE INFINITELY LARGE*

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

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-

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

*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.

Dictionary of the History of Ideas | ||