VII. THE COMPLETE AND THE PERFECT
Nonscientific aspects of infinity are usually broad and
elusive and mottled with ambiguities and polarities.
One of the worst offenders was Benedict Spinoza,
however much he presumed to articulate his thoughts
more geometrico. In fact, the term “infinite” stands in
Spinoza for such terms as “unique,” “incomparable,”
“homonymous,” “indeterminate,” “incomprehensible,”
“ineffable,” “indefinable,” “unknowable,” and many
other similar terms (Wolfson,... Spinoza, I, 138).
What is worse, Spinoza justified this license of his by
reference to Aristotle's dictum that “the infinite so far
as infinite is unknown” (ibid., I, 139), which Aristotle
certainly would not have allowed to be exploited in
this way.
But even when intended to be much more coherent,
the conception of infinity in a nonscientific context,
especially in theology, need not refer to the magnitude
of quantitative elements like space, time, matter, etc.,
but it may refer to the intensity of qualitative attributes
like power, being, intellect, justice, goodness, grace,
etc. There are large-scale philosophical settings, in
which infinity, under this or an equivalent name, does
not magnify, or even emphasize, the outward extent
of something quantifiable, but expresses a degree of
completeness and perfection of something structurable.
Because of all that, philosopher-theologians who
strive for clarity of thought and exposition are having
great difficulties with them. Thus, Saint Thomas
Aquinas, in a discourse on the existence and nature
of God in the entering part of his Summa theologiae,
compares and confronts the completeness and perfec-
tion in God with the infinite and limitless in Him. In
a “typically Thomistic” sequence of arguments and
counterarguments, completeness and infinity are al-
ternately identified and contrasted, as if they were
synonyms and antonyms in one; and, although Aquinas
very much strives for clarity, it would be difficult to
state in a few sharply worded declaratory statements,
what the outcome of the discourse actually is (Saint
Thomas Aquinas, Summa theologiae, Vol. II).
Completeness in philosophy is even harder to define
than infinity in philosophy, and the relation between
the two is recondite and elusive. The problem of this
relation was already known to the Greeks. As a prob-
lem of cognition it was created by Parmenides, and
then clearly formulated by Aristotle, but as a problem
of “systematic” theology it came to the fore only in
the second half of Hellenism, beginning recognizably
with Philo of Alexandria, and coming to a first culmi-
nation in the Enneads of Plotinus. From our retrospect,
the “One” (τὸ ἕν) of Plotinus was a fusion between a
divinely intuited completeness and a metaphysically
perceived infinity. Books V and VI of the Enneads are
full of evidence for this, and we note, for instance, that
a recent study of Plotinus summarizes the passage VI,
8.11, of the Enneads thus:
The absolute transcendence of the One as unconditioned,
unlimited, Principle of all things: particular necessity of
eliminating all spatial ideas from our thoughts about Him
(A. H. Armstrong, Plotinus, p. 63).
Also, a study of Plotinus of very recent date has the
following important summary:
Within recent years there has been a long and learned
discussion on the infinity of the Plotinian One, and from
it we learn much. The chief participants are now in basic
agreement that the One is infinite in itself as well as infinite
in power
(J. M. Rist, p. 25).
Long before that, Aristotle devoted a chapter of his
Physica (Book 3, Ch. 6) to an express comparison be-
tween completeness and infinity, as he saw it. Aristotle
presents a thesis that infinity is directly and unmistaka-
bly opposed to “the Complete and the Whole” (τέλειον
καὶ ὅλον), and his central statement runs as follows:
The infinite turns out to be the contrary of what it is said
to be. It is not what has nothing outside it that is infinite,
but what always has something outside it
(206b 34-207a 1, Oxford translation).
His definition then is as follows:
A quantity is infinite if it is such that we can always take
a part outside what has already been taken. On the other
hand what has nothing outside it is complete and whole.
For thus we define the whole—that from which nothing
is wanting—as a whole man or a whole box
(ibid., 207a 7-11).
'Whole' and 'complete' are either quite identical or closely
akin. Nothing is complete (teleion) which has no end (telos);
and the end is a limit
(ibid., 207a 13-14).
Immediately following this passage, Aristotle makes
respectful mention of Parmenides, and deservedly so.
The great ontological poem of Parmenides clearly
outlines a certain feature of completeness, as an attri-
bute of something that is, ambivalently, an ontological
absolute and a cosmological universe. Ontologically
this universe was made of pure being and thought
itself, and there has been nothing like it since then
(W. K. C. Guthrie, A History..., Vol. 2; Untersteiner,
Parmenide...; L. Tarán, Parmenides...). And yet,
as we have tried to demonstrate in another context,
the Parmenidean completeness was so rich in allusions
that it even allows a measure of mathematization in
terms of today, more so than Aristotle's interpretation
of this completeness would (Bochner, “The Size of the
Universe...,” sec. V).
The Parmenidean being and thought, as constituents
of the universe, were conceived very tightly. In the
course of many centuries after Parmenides, they were
loosened up and gradually transformed into the
Hellenistic “One” and “Logos,” which were conceived
more diffusely, and less controversially. Also, in the
course of these and later centuries, the Parmenidean
universe, with its attribute of completeness, was overtly
theologized, mainly Christianized.
Aristotle took it for granted that the ontological
universe of Parmenides, in addition to being complete,
was also finite, and Parmenides did indeed so envisage
it, more or less. But what was a vision in Parmenides
was turned into a compulsion by Aristotle. That is,
Aristotle maintained, and made into a major proposi-
tion, that the Parmenidean universe could not be other
than finite, because, for Aristotle, completeness some-
how had to be anti-infinite automatically.
With this proposition Aristotle may have over-
reached himself. Mathematics has introduced, entirely
from its own spontaneity, and under various names,
several versions of completeness, any of which is remi-
niscent of the notion of Parmenides, and, on the whole,
finiteness is not implied automatically. On the contrary,
the completeness of Parmenides can be mathematically
so formalized, that a universe becomes complete if it
is so very infinite that no kind of magnification of it
is possible (Bochner, loc. cit.). But mathematizations
of the conception of completeness are of relatively
recent origin, and it would not be meaningful to pursue
the comparison between mathematical and philo-
sophical versions of the conception beyond a certain
point.
