Chapter 49
The infinite and the finite; the unbounded, the endless and the infinitely small or great.
113 passages · 38 principal · Covers the 49 volumes of primary texts
Few ideas run so evenly through the Harvard Classics, or gather such different kinds of writers, as the infinite. Mathematicians and astronomers ask whether quantity can be divided without end and whether the heavens have a boundary. Theologians and the authors of sacred books name the divine by its lack of limit. Moralists measure a human life against boundless time. Critics and poets find in the endless the source of the sublime and the object of longing. Under all these inquiries lies a single question that is never settled: whether a finite mind can know, or even coherently think, what has no bounds. That question touches mathematics, cosmology, religion and art at once, and these books answer it in sharply different ways. This is why the idea deserves a place among the great ones.
The most austere answer is Hobbes’s. Whatever we imagine is finite, he holds, so the word “infinite” names no conception of a thing; it records only that we cannot conceive its bounds1. Berkeley, though no friend of Hobbes, grants the premise in part. Sense and imagination cannot take in the boundless extent of the universe, and an immeasurable surplus always remains beyond them2. Against this, a larger company holds that the infinite is known to exist even though its nature escapes us. Pascal states the position most exactly. We know there is an infinity of number, yet we cannot say whether it is odd or even3. We know the infinite exists but not what it is, which is the reverse of our case with the finite4. Infinite number, or an infinite space equal to a finite one, are things that are incomprehensible and yet real5, and an infinity of things lies beyond reason altogether6. Dante had already found the number of the angelic orders beyond mortal counting, the finite number in Daniel concealing an infinite7. Rousseau’s vicar takes a more cautious line. The unlimited lies beyond his comprehension, so he will neither affirm nor deny it8, and the longer he contemplates God’s infinite essence, the less he can conceive it9. Carlyle gives the confession a Romantic cast. Man’s handbreadth of space is ringed by Infinitude10. The finite can never render a complete theorem of the infinite11, and its mystery is read only in fragments12.
Kelvin turns the whole debate around. For him it is finitude that cannot be comprehended. A universe with an end cannot be conceived, while the infinity of space and matter can13. Set beside Hobbes, the reversal is striking. One thinker finds the boundary unthinkable, the other the boundlessness, and the disagreement shows how far the answer depends on whether one starts from the image or from the concept.
The mathematicians press the question hardest, because they must reason with infinities whether they comprehend them or not. Schiller names the analysis of the infinite as a typical product of specialised abstraction14. Voltaire reports with admiration that the geometers have subjected infinity to calculation15. He grants that Newton’s fluxions, with their infinite orders of infinites, look like frenzy, yet he counts them truths of human subtlety, of a piece with infinite divisibility16. He extends the same reasoning to attraction, which holds in the half, the quarter and so on in infinitum17.
Divisibility is where the controversy is sharpest. Pascal argues that geometers must hold space to be infinitely divisible, though no one comprehends infinite division, and he notes that some deny it simply because it cannot be conceived18. His reply rests on a principle of method. Both alternatives are inconceivable, yet one of them must be true19. The paradox of motion dissolves once we see that an infinity of divisible parts is crossed in an infinity of moments, and that a small space contains as many parts as a great one20. Hume answers in the opposite spirit. He sets out the infinite divisibility of extension and the infinitely small angles of contact as absurdities that the geometers nonetheless demonstrate21. He judges the infinite divisibility of time still more plainly absurd22, and he concludes that ideas of quantity are not infinitely divisible, which disposes of the paradoxes23. Berkeley also traces the paradoxes to the assumption that finite extension is infinitely divisible24. In the same spirit he rejects a material substratum, which would need an extension of its own and so begin an absurd regress25. Pascal and Hume therefore agree that infinite division is inconceivable. They part over whether inconceivability counts as refutation.
From divisibility Pascal builds his general doctrine of the two infinitudes. Every motion, number and space has an infinity of greatness and an infinity of littleness, so that it can always be exceeded or reduced26. Any magnitude can be doubled or halved without end, and so it stands between nothingness and the infinite27. Infinite increase implies infinite division28. These two infinities are set before man to be admired, not comprehended29, and Pascal defends the length of his treatment for readers who still need persuading30. The repetition of spaces and numbers yields a kind of infinity and eternity, although strictly only the number that multiplies them is infinite31. All finite quantities, Pascal adds, are equal when set against the Infinites32, since the finite lies between two infinites that enclose it and flee from it33. Other writers use the word more loosely. Adam Smith calls the possible extent of the carrying trade “in a manner” infinite34. Burke notes that every subject branches out without end35. Darwin, for his part, refuses to declare variation exhausted while still admitting a natural limit, such as the limit friction and muscle set to an animal’s speed36.
Whether the universe itself is bounded the texts leave unresolved. The Buddhist writings list the finitude or infinitude of the world among the theories the Buddha left unelucidated37, even while they describe meditative realms of infinite space and infinite consciousness38. Milton’s Deep is illimitable and without bound39. Newcomb argues on observational grounds that the universe is not infinite. If it were, the sky would blaze with countless stars, and in fact the star counts fall off40. Yet he admits that no definite limit can be fixed, so what lies beyond remains an open question41. His caution sits between Kelvin’s confidence and the Buddhist suspension of judgement.
What the endless cosmos means for man is Pascal’s great theme. The universe is an infinite sphere whose centre is everywhere, and imagination tires before nature does42. The smallest mite opens an abyss of littleness to match the abyss of greatness43, with universes within universes repeated without end44. A landscape seen up close dissolves into houses, trees, leaves, ants and the limbs of ants, in infinity45. Man hangs between the Infinite and Nothing and cannot grasp either extreme46. Knowing the Nothing would take as infinite a capacity as knowing the All, and the extremes meet only in God47. All things proceed from Nothing toward the Infinite48. Man sees infinites on every side and is himself an atom and a passing shadow49. The eternal silence of those infinite spaces frightens him50. This vision is meant to humble pride51,52, and Pascal finds even in the sciences an infinity of problems and premises53. Sainte-Beuve sees the same image of kingdoms as specks in the infinite foreshadowed in Montaigne54. Darwin, a naturalist rather than a moralist, marvels instead of trembling at creatures beyond counting in a drop of water55 and at confervae in the sea that must be infinite in number56.
Endless time raises its own reflection, and here the ancients speak most directly. Marcus Aurelius sees a chaos of infinite time on either side of the present57. Against boundless time before and after, three days and three generations come to the same58. It makes no difference whether one watches the same things for a hundred years or for infinite time59, and causes work in endless variety across a boundless past and future60. For the Stoic this vastness is a remedy for anxiety, whereas for Pascal it is a source of fear. Darwin turns boundless time to a scientific use. He asks whether any power acting for less than an infinite time could have stripped so vast an expanse of granite61, and he likens the great multiplier of a repeated cause to a savage pointing at the hairs of his head62.
In theology the infinite is first of all the name of God. The Bhagavad-Gita calls the Unmanifest the Infinite and the All63, a form without end, beginning or centre, present everywhere in space64. Augustine reasons that a bodily infinite would be less in a part than in the whole, whereas God as spirit is wholly everywhere65. He describes his earlier, mistaken way of picturing God as boundless space containing all things66, and he sets the finite creation within God like a sponge within a boundless sea67. Penn likewise speaks of God as infinite Spirit present everywhere at once68. Raleigh argues from infinite power to the world’s beginning69. Infinite matter, he holds, would leave no room for form, and God’s power is limited only by his will70. Because the finite world bears no proportion to the infinite, its coming from the infinite is no wonder71, and the divine virtue that upholds it must itself be infinite, though the world receives it gradually72. Rousseau’s vicar denies that causes can run back to infinity, since that would amount to having no first cause73. Milton’s God confines vast Infinitude at his bidding74, yet Adam asks whether an infinite wrath can be drawn out upon finite man75. For Lessing, reason derives its true conception of the One from the Infinite One76. Wordsworth makes infinitude the element of religion, in which the Infinite Being accommodates himself to finite capacity77. Pascal sets the finite self against an infinite Being78, describes a soul that contemplates God through immensities it multiplies without ceasing79, and calls Christ’s model of perfection infinite where the world’s is limited80.
Schiller adds a philosophical refinement. Mere unlimited determinability is an “empty infiniteness”, and reality arises only by limiting infinite space and time81. The finite mind reaches the absolute only through limitation82. Aesthetic freedom, by contrast, is a “completed infiniteness”83. Carlyle and Channing reverse the emphasis and see the finite as the infinite made manifest. Nature bodies forth the Finite from the Infinite84, and the known harmonizes with the boundless unknown85.
Pascal’s wager turns the arithmetic of infinity to practical use. The finite is annihilated before the infinite, and a unit added to infinity adds nothing86. An infinity of infinitely happy life therefore outweighs any finite stake87. Since the distance between certainty and uncertainty is not infinite, it is reasonable to risk everything88, and the proposition acquires infinite force89. Here the two infinitudes of his geometry become a calculation about salvation.
Burke carries the idea into aesthetics. Eternity and infinity are among the most affecting of ideas, though we understand them least90. To see bounds distinctly is to diminish a thing, so a clear idea is a little idea91. Infinity fills the mind with a “delightful horror”, which Burke calls the truest test of the sublime, and the eye takes unbounded things as infinite92. Succession and uniformity of parts produce an artificial infinite by giving the imagination no check93,94. A colonnade therefore gives infinity, where a bare wall gives only vastness95, and a grander effect comes from less matter96. The multitude of the stars gives the same appearance97. Infinity can also please, through the promise of something more98. Burke agrees with Pascal that the imagination is lost as much in the minute as in the vast99, though he holds that small parts cannot raise it to infinity100. Dante’s sight trembles at the brink of dread infinitude101, and Darwin finds that boundless plains stir ill-defined feelings102.
Finally the infinite becomes the goal of striving. Plutarch’s Cicero reflects that glory is an infinite thing with no fixed measure103. Faust laments that he is no nearer the infinite104. Schiller places the ideal as an infinite that man approaches without reaching105, into which he extends his individuality106. Emerson’s circles are always surrounded by further circles107, and the infinite hope of genius degrades whatever it has achieved108. Renan finds the Celtic imagination to be the infinite set against the finite, pursued even beyond the tomb109. Shelley calls high poetry infinite, like an acorn holding all oaks110, and Blake holds infinity in the palm of the hand111.
The questions remain open. These books do not settle whether the infinite is a positive object or a confession of limits, whether divisibility is a truth or an absurdity, whether the cosmos is bounded, or whether the endless calls for Stoic calm, Pascalian dread or Romantic aspiration.
Introductory essay written by Claude Opus 5.5 from the outline and the notes on every passage below; quotations are checked against this edition.
See also: The Sublime 2c
See also: Space, Eternity, The Sublime, Mathematics, God, World