INFINITY
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INFINITY
the unity of God, the grand thought of TertuUian
(Adv. Marcion., I, iii), " If God is not one, He is not at
all." But that besides God there are creatures of His,
reflections from His light, illumined only by Him and
in no way diminishing His light, docs not limit God
Himself. God, on the contrary, would be finite, if His
creatures were identical with Him. For creatures are
essentially of mixed perfection, because essentially de-
pendent; infinite is only that which is pure perfection
without any admixture of imperfection. If, there-
fore, one wants to form the equation: infinite=all, it
must be interpreted: infinite = everything uncreated;
or better still: infinite =all pure perfections in the
highest and truest sense. Taken in the monistic view,
viz. that there can be no reality besides the infinite, this
equation is wrong. The identification, however, of
"infinite" and "all" is very old, and served as a
basis of the Eleatic philosophy.
Another very common objection of Monists against the theistic conception of God is, that being personal, He cannot be infinite. For personality, whether conceived as individuality or as self-consciousness or as subsistent being, cannot exist without something else as its opposite ; but, wherever there is something else, there is no infinity. Both premises of this argu- ment are false. To assert that infinity is destroyed wherever something else exi.sts, is but the repetition of the already rejected statement that infinity means totality. Equally unwarranted is the assertion that personality requires the existence of something else. Individuality means nothing more than that a thing is this one thing and not another thing, and it is just as much this one thing, whether anything else exist or not. The same is true of self-consciousness. I am aware of myself as Ego, even though nothing else exist, and though I have no thought of any other being; for the Ego is something absolute, not relative. Only if I desire to know myself as not being the non-Ego — to use the expression of Fichte — I necessarily must think of that non-Ego, i. e. of some- thing as not-myself. The subsistence of intellectual beings, i. e. personality in the strictest sense of the term, implies only that I am a being in and for myself, separate from everything else and in no way part of anything else. This would be true, even though nothing else existed; in fact, it would then be truer than ever. Far from excluding personality God is personal in the deepest and truest meaning, because He is the most independent Being, by Himself and in Himself in the most absolute sense (see Pekson).
History. — Concerning the philosophers before Aristotle, Suarez pertinently remarks that they "scented" the infinity of God {subodorati sunt). In many of them we meet the infinity of God or of the First Cause, though in many cases it be only infinity in extension. Plato and Aristotle assert in substance the infinity of the Highest Being in a more adequate sense, though blended with errors and obscur- ities. The Stoics had various ideas that would have led them to admit the infinity of God, had not their Pantheism stood in the way. The conceptions of Philo's Jewish-Alexandrian philosophy were much purer ; the same may be said to a certain degree of the neo-Platonism of Plotinus, who was largely influenced by Philo. Plotinus originated the terse and trenchant argument: God is not limited; for what should limit Him? (" Enn. V", lib. V, in " Opera omnia", Oxford, 1885, p. 979). Against Plotinus, however, it may be objected that true infinity is as little consistent with his doctrine of emanations as with the rnore or less pantheistic tendencies of the Indian philos- ophy.
The Christian writers took their concepts of the infinity of God from the Bible; the speculative devel- opment of these ideas, however, needed time. St. Augustine, Iseing well acquainted with Platonic philo.so- phy, recognized that whatever could be greater, could
not be the First Being. Candidus, a contemporary
of Charlemagne, perceived that the limitations of all
finite beings point towards a Creator, Who deter-
mines the degrees of their perfection. Abelard seems
to teach that God, being superior to everything else in
the reason of His existence, must also be greater in His
perfections. A book, which is sometimes ascribed to
Albert the Great, derives God's infinity from His pure
actuality. All these reasons were collected, devel-
oped, and deepened by the Scholastics of the best
period; and since then the speculative proof for the
infinity of God has, in spite of some few objectors,
been considered as secure. Even Moses Mendelssohn
writes: "That the necessary Being contains every
perfection which it has, in the highest possible degree
and without any limitations, is developed in nvmiber-
less text-books, and so far nobody has brought a
serious objection against it" (" Gesanimelte Schrif-
ten", II, Leipzig, 189.3, p. .355). Kant's attempt to
stigmatize the deduction of infinity from self-exist-
ence as a return to the ontological argument, was a
failure; for our deduction starts from the actually
existing God, not from mere ideas, as the ontological
argument does. Among Christians, the dogma itself
has been rarely denied, but the freer tendencies of
modern Protestantism in the direction of Pantheism,
and the views of some champions of Modernism in
the Catholic Church, are in fact, though not always in
expression, opposed to the infinity of God.
Infinity of Creatures. — The knowledge we have about the infinity of creatures leaves much to be desired. It is certain that no creature is infinite in every regard. However great it may be, it lacks the most essential perfection: self-existence, and what- ever is necessarily connected with it. Moreover, philosophers and theologians are practically unani- mous in declaring that no creature can be infinite in an essential predicate. As to the questions whether an accident (e. g. quantity) is capable of infinity, whether the creation could lie infinite in extension, whether there can be an infinite number of actual beings, or whether an infinite number is at all possible — as to these questions they are less in harmony, though the maj ority lean towards the negative answer, and in our time this majority seems to have increased. At any rate the infinite world, of which the old Greek philosophers dreamt and the moilern Materialists and Monists talk so much, lacks every proof, and, as to the infinite duration of the world, it is contradicted by the dogma of its temporal beginning.
The mathematicians too occupy themselves with the infinite, both with the infinitely small and the infinitely large, in the treatises on infinite series, and infinitesimal calculus, and generally in all limit opera- tions. The infinitely small is represented by the sign 0, the infinitely large by oo, and their relation is expressed by the ratio ^ = co . All mathematicians agree as to the method of operating with the two quantities; but there is much division amongst philos- ophers and philosophizing mathcnuiticiaiis as to their real meaning. The least subject to difiiculties are perhaps the following two views. The infinite in mathematics may be taken as the potentially infinite, i. e. that which can be increased or diminishcil with- out end; in this view it is a real quantity, cajiable of existence. Or one may take it as the actually infinite, viz. that which by actual successive addition or divi- sion can never be reached. In this view it is some- thing which can never exist in reality, or from the possibility of whose existence we at best abstract. It is a limit which exists only as a fiction of the mind (ens rationis). Or if the infinitely small is considered as an absolute zero, but connoting dif- ferent values, it is really a limit, liut as far as it connotes other values, only a logical being. Thus, at times Leibniz calls both the infinitely small and the infinitely large fictions of the
