(Latin infinitas; in, not, finis , the end, the boundary).
The infinite, as the word indicates, is that which has no end, no limit, no boundary, and therefore cannot be measured by a finite standard, however often applied; it is that which cannot be attained by successive addition, not exhausted by successive subtraction of finite quantities. Though in itself a negative term, infinity has a very positive meaning. Since it denies all bounds -- which are themselves negations -- it is a double negation, hence an affirmation, and expresses positively the highest unsurpassable reality. Like the concepts of quantity, limit, boundary, the term infinity applies primarily to space and time, but not exclusively, as Schopenhauer maintains. In a derived meaning it may be applied to every kind of perfection: wisdom, beauty, power, the fullness of being itself.
The concept of infinity must be carefully distinguished from the concept of "all-being". Infinity implies that an infinite being cannot lack any reality in the line in which it is infinite, and that it cannot be surpassed by anything else in that particular perfection; but this does not necessarily mean that no other being can have perfections. "All-being", however, implies that there is no reality outside of itself, that beyond it there is nothing good, pure, and beautiful. The infinite is equivalent to all other things put together; it is the greatest and most beautiful; but besides it, other things both beautiful and good may exist (for further explanation, see below). It is objected that, if there were an infinite body, no other body could exist besides it; for the infinite body would occupy all space. But the fact that no other body could exist besides the infinite body would be the result of its impenetrability, not of its infinity. Spinoza defines: "Finite in its kind is that which can be limited by a thing of the same kind." (Ethics, I def. ii). If he intended only to say: "Finite is that from which another thing of the same kind, by its very existence, takes away perfection", no fault could be found with him. But what he means to say is this: "Finite is that, besides which something else can exist; infinite therefore is that only which includes all things in itself." This definition is false.
Many confound the infinite with the indeterminate. Determination (determinatio) is negation, limitation (negatio, limitatio) , says Spinoza. Generally speaking, this is false. Determination is limitation in those cases only where it excludes any further possible perfection, as for example, the determination of a surface by a geometrical figure; but it is no limitation, if it adds further reality, and does not exclude, but rather requires a new perfection, as for example, the determination of substance by rationality. The mere abstract being, so well known to metaphysicians, is the most indeterminate of all ideas, and nevertheless the poorest in content; the infinite, however, is in every way the most determinate idea, in which all possibilities are realized, and which is therefore the richest in content. According to Hobbs, we call a thing infinite if we cannot assign limits to it. This definition is also insufficient: infinite is not that whose limits we cannot perceive, but that which has no limit.
The different kinds of infinity must be carefully distinguished. The two principal divisions are: (1) the infinite in only one respect (secundum quid) or the partially infinite, and the infinite in every respect (simpliciter) or the absolutely infinite; (2) the actually infinite, and the potentially infinite, which is capable of an indefinite increase. Infinite in only one respect (viz. extension ) is ideal space ; infinite in only one respect (viz. duration) is the immortal soul ; infinite in every respect is that being alone, which contains in itself all possible perfections and which is above every species and genus and order. Potentially infinite is (e.g.) the path of a body which moves in free space ; potentially infinite is also the duration of matter and energy, according to the law of their conservation . For this motion and this duration will never cease, and in this sense will be without end; nevertheless, the path and the duration up to this instant can be measured at any given point and are therefore in this sense finite. Hence they are infinite not according to what they actually are at a given moment, but according to what they are not yet and never actually can be; they are infinite in this, that they are ever and forever progressing without bounds, that there is always the "and so forth". The actually infinite, however, is now and at every moment complete, absolute, entirely determined. The immeasurable omnipresent spirit does not advance from point to point without end, but is constantly everywhere, fills every "beyond" of every assignable point. Hegel calls potential infinity the improper (schlechte) , actual infinity the true infinity.
The actual infinity of God in every respect is Catholic dogma. In accordance with the Holy Bible ( 1 Kings 8:27 ; Psalm 144:3 ; 146:5 ; Sirach 43:29 sqq. , Luke 1:37 , etc.) and unanimous tradition, the Vatican Council at its Third Session (cap. i) declared God to be almighty, eternal, immense, incomprehensible, infinite in intellect and will and every perfection, really and essentially distinct from the world, infinitely blessed in Himself and through Himself, and inexpressibly above all things that can exist and be thought of besides Him. The infinity of God can also be proved from philosophy. God is the self-existing, uncreated Being whose entire explanation must be in Himself, in Whom there can be no trace of chance; but it would be mere chance if God possessed only a finite degree of perfection, for however high that degree might be, everything in the uncreated Being -- His perfections, His individuality, His personality -- admit the possibility of His possessing a still higher degree of entirety. From outside Himself, God cannot be limited, because, being uncreated, He is absolutely independent of external causes and conditions. Limitation would be chance; the more so because we can maintain not only that any given finite degree of perfection may be surpassed, but also, in a positive way, that an infinite being is possible. Moreover, if God were finite, the existence of other gods, His equals or even His superiors in perfection would be possible, and it would be mere chance if the y did not exist. Of such gods, no trace can be found, while on the other hand, God's infinity is suggested by various data of experience, and in particular by our unbounded longing after knowledge and happiness. The more man a man is, and the more he follows his best thoughts and impulses, the less he is satisfied with merely finite cognitions and pleasures. That the essential cravings of our nature are not deceptive, is demonstrated at once by experience and speculation.
From the infinity of God it is easy to deduce all His perfections: His unity, simplicity, immutability, etc., though these may be proved also by other means. Many of God's attributes are nothing else than His infinity in a particular respect, e.g. His omnipotence is but the infinity of His power; His omniscience, the infinity of His knowledge. Whatever is known to be a pure unalloyed perfection, must be an attribute of God on account of His infinity. We say a pure unalloyed perfection; for God, just because He is infinite does not possess all perfections in the same way. Only pure perfections -- i.e. those which include in their concept no trace of imperfection whatsoever -- are contained in Him formally. We must therefore ascribe to Him the attributes wise, powerful, amiable, etc., without any restriction, because these are all pure perfections. Of the so-called mixed perfections, which include besides the positive reality also some imperfections, as e.g., extension, contrition, courage, sound reasoning, and clear judgment, He possesses only the perfection without the connected imperfection. His is, for example, the all-pervading presence, without composition; love for the good without having committed sin ; power without having to overcome fear ; knowledge without formal reasoning or formal judgment. He possesses, therefore, the mixed perfections in a higher form -- eminently, i.e. in the only form which is worthy of the infinite. But even the pure perfections are contained in Him in a higher form than in the creature, in which they are dependent, derived, finite. God's perfection and that of the creature are the same analogically only, not univocally. The error of Anthropomorphism consists just in this, that it ascribes to God human perfections, without first refining them; whereas Agnosticism errs in its contention that, of all the pure and good qualities which are found in creatures, none can be ascribed to God. Those modern writers too are mistaken, who hold the best form of religious sentiment to be that which comprises the largest number of elements, and if needs be, of contradictions. According to them, we should call God both finite and infinite; finite to escape Agnosticism, infinite to escape Anthropomorphism. But it is evident that the highest and absolute truth cannot be a compound of contradictions.
The dogma of God's infinity is not only of the greatest import for theology in the strictest sense of the term (i.e. the treatise on God ), but it throws new light upon the malice of sin, which, on the account of Him Who is offended, becomes objectively infinite; upon the infinite majesty of the Incarnate Word and the boundless value of His merits and satisfaction; upon the necessity of the Incarnation, if God's justice required an adequate satisfaction for sin.
How imperatively thought demands that infinity be ascribed to the self-existent Being is best shown by the fact, that all those who have at any time identified, and especially those who nowadays identify God and the world -- in short, all Monists -- almost universally speak if the infinity of their God. But this is an error. One has but to open one's eyes to see that the world in imperfect, and therefore finite. It avails nothing for the Monists to assume that the world is infinite in extension ; all that could be inferred from this supposition would be an infinitely extended imperfection and finiteness. Nor do they gain anything by staking their hopes on evolution, and predicting infinity for the future of the world; uncreated existence involves infinity at every moment, at this present instant as well as at any future time, and not only potential but real, actual infinity. Others therefore maintain that the world is not their God, but an emanation from God ; they must consequently grant that God has parts -- else nothing could emanate from Him -- and that these parts are subject to imperfections, decay, and evil -- in short that their God is not infinite. Hence others assert that the things of this world are not parts of the Absolute, but its manifestations, representations, forms, qualities, activities, accidents, attributes, affections, phenomena, modifications. But if these are not mere words, if the things of this world are really modifications etc. of the Absolute, it follows again that, as much as it is in finite things, the Absolute is subject to limitation, evil and sin, and is therefore not infinite. This leads many to take the last step by asserting that the things of this world are nothing in them selves, but simply thoughts and conations of the Absolute. But why has not the Absolute grander and purer conceptions and volitions? Why has it contented itself for thousands of years with these realistic self-represent ations, and not even yet attained with certainty an idealistic conception of reality? Turn as one may, in spite of all efforts to evade the consequence, the god of Monism is not an infinite being.
The Monists object that God as conceived by Theists is a finite thing, since He is not in Himself all reality, but has outside Himself, the reality of the world. However, it has been stated above that infinity and totality are two entirely different ideas, and that infinity does not, in every supposition, exclude the existence of other things besides itself. We say, not "in every supposition", for it may be that the infinite could not be infinite if certain beings existed. A being uncreated or independent of God, or a Manichæan principle of evil, cannot exist beside the infinite God, because it would limit His absolute perfections. This is the time-honored proof for the unity of God, the grand thought of Tertullian (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, illuminated only by Him and in no way diminishing His light, does 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 dependent; infinite is only that which is pure perfection without any admixture of imperfection. If, therefore, 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, how ever, of "infinite" and "all" is very old, and served as a basis of 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 argument are false. To assert that infinity is destroyed wherever something else exists, 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 not hing 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 exists or not. The same is true of self-consciousness. I am aware of myself as Ego, even though nothing else exist, and 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 something 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 PERSON).
Concerning the philosophers before Aristotle, Francisco Suárez 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 obscurities. 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 more or less pantheistic tendencies of the Indian philosophy.
The Christian writers took their concepts of the infinity of God from the Bible ; the speculative development of these ideas, however, needed time. St. Augustine, being well acquainted with Platonic philosophy, 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 determines 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, developed, 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 numberless text-books, and so far nobody has brought a serious objection against it" ("Gesammelte Schriften", II, Leipzig, 1893, p. 355). Kant's attempt to stigmatize the deduction of infinity from self-existence 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 rarely been 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, although not always in expression, opposed to the infinity of God.
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 whatever else is necessarily connected with it. Moreover, philosophers and theologians are practically unanimous 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 be 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 majority lean towards the negative answer, and in our time this number seems to have increased. At any rate the infinite world, of which the old Greek philosophers dreamt and the modern 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 operations. The infinitely small is represented by the sign 0, the infinitely large by a character that looks like the number "8" turned on its side. Their relation is expressed by the ratio 1/0 = (infitely large) All mathematiians agree as to the method of operating with the two quantities; but there is much division amongst philosophers and philosophizing mathematicians as to their real meaning. The least subject to difficulties are perhaps the following two views. The infinite in mathematics may be take n as the potentially infinite, i.e. that which can be increased or diminished without end; in this view it is a real quantity, capable of existence. Or one may take it as the actually infinite, viz. that which by actual successive addition or division can never be reached. In this view it is something 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 different values, it is really a limit, but 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 mind ( mentis fictiones ) and compares them to imaginary quantities. Carnot calls the differential an être de raison ; Gauss speaks of a façon de parler .
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