“The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.”

Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Context: This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers... Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
... The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
... to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.<!--pp.10-12

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formul…" by Henri Poincaré?
Henri Poincaré photo
Henri Poincaré 49
French mathematician, physicist, engineer, and philosopher … 1854–1912

Related quotes

Marshall McLuhan photo

“Formal logic and the logical syllogism encapsulate connectedness in reasoning.”

Marshall McLuhan (1911–1980) Canadian educator, philosopher, and scholar-- a professor of English literature, a literary critic, and a …

1980s, Laws of Media: The New Science (with Eric McLuhan) (1988)

Grant Morrison photo

“The interior of our skulls contains a portal to infinity.”

Grant Morrison (1960) writer

Source: Supergods: What Masked Vigilantes, Miraculous Mutants, and a Sun God from Smallville Can Teach Us About Being Human

Oliver Hazard Perry photo

“Of Captain Elliot, already so well known to the government, it would be almost superfluous to speak; in this action, he evinced his characteristic bravery and judgment; and, since the close of the action, has given me the most able and essential assistance.”

Oliver Hazard Perry (1785–1819) United States Naval Officer

Report on the Battle of Lake Erie (13 September 1813); Years later Perry would declare he had sought to minimize what he perceived to be a lack of valor on the part of Elliot, and requested a court-martial against him, for this and other matters.

Lev Vygotsky photo

“As in the focus of a magnifying glass, play contains all developmental tendencies in a condensed form and is itself a major source of development.”

Lev Vygotsky (1896–1934) Soviet psychologist

Vygotsky, L. S. (1930) Mind in Society: The Development of Higher Psychological Processes Cambridge, Mass.: Harvard University Press p.102

George Sand photo

“He made a single instrument speak a language of infinity. He could often sum up, in ten lines that a child could play, poems of a boundless exaltation, dramas of unequalled power.”

George Sand (1804–1876) French novelist and memoirist; pseudonym of Lucile Aurore Dupin

On Chopin's Preludes in Histoire de Ma Vie (1902-04), Vo. IV, p. 439
Context: It was there he composed these most beautiful of short pages which he modestly entitled the Preludes. They are masterpieces. Several bring to mind visions of deceased monks and the sound of funeral chants; others are melancholy and fragrant; they came to him in times of sun and health, in the clamor of laughing children under he window, the faraway sound of guitars, birdsongs from the moist leaves, in the sight of the small pale roses coming in bloom on the snow. … Still others are of a mournful sadness, and while charming your ear, they break your heart. There is one that came to him through an evening of dismal rain — it casts the soul into a terrible dejection. Maurice and I had left him in good health one morning to go shopping in Palma for things we needed at out "encampment." The rain came in overflowing torrents. We made three leagues in six hours, only to return in the middle of a flood. We got back in absolute dark, shoeless, having been abandoned by our driver to cross unheard of perils. We hurried, knowing how our sick one would worry. Indeed he had, but now was as though congealed in a kind of quiet desperation, and, weeping, he was playing his wonderful Prelude. Seeing us come in, he got up with a cry, then said with a bewildered air and a strange tone, "Ah, I was sure that you were dead." When he recovered his spirits and saw the state we were in, he was ill, picturing the dangers we had been through, but he confessed to me that while waiting for us he had seen it all in a dream, and no longer distinguished the dream from reality, he became calm and drowsy while playing the piano, persuaded that he was dead himself. He saw himself drowned in a lake. Heavy drops of icy water fell in a regular rhythm on his breast, and when I made him listen to the sound of the drops of water indeed falling in rhythm on the roof, he denied having heard it. He was even angry that I should intepret this in terms of imitative sounds. He protested with all his might — and he was right to — against the childishness of such aural imitations. His genius was filled with the mysterious sounds of nature, but transformed into sublime equivalents in musical thought, and not through slavish imitation of the actual external sounds. His composition of that night was surely filled with raindrops, resounding clearly on the tiles of the Charterhouse, but it had been transformed in his imagination and in his song into tears falling upon his heart from the sky. … The gift of Chopin is [the expression of] the deepest and fullest feelings and emotions that have ever existed. He made a single instrument speak a language of infinity. He could often sum up, in ten lines that a child could play, poems of a boundless exaltation, dramas of unequalled power.

Gustave Flaubert photo
Kurt Lewin photo

“The essential meaning of such an assertion is this: events a and b are necessarily dependent moments of a single unified occurrence. The mathematical formula states the quantitative relations involved in the occurrence. Already in such cases the dependent moment of the occurrence are moments that obtain temporally by side.”

Kurt Lewin (1890–1947) German-American psychologist

Kurt Lewin (1927, p. 305) as cited in: K. Mulligan & B. Smith (1988) " Mach and Ehrenfels: Foundations of Gestalt Theory http://ontology.buffalo.edu/smith/articles/mach/mach.pdf". p. 149.
1920s

Frank Herbert photo

“No matter how finely you subdivide time and space, each tiny division contains infinity.”

Frank Herbert (1920–1986) American writer

Dune Genesis (1980)
Context: No matter how finely you subdivide time and space, each tiny division contains infinity.
But this could imply that you can cut across linear time, open it like a ripe fruit, and see consequential connections. You could be prescient, predict accurately. Predestination and paradox once more.
The flaw must lie in our methods of description, in languages, in social networks of meaning, in moral structures, and in philosophies and religions — all of which convey implicit limits where no limits exist. Paul Muad'Dib, after all, says this time after time throughout Dune.

Marsilio Ficino photo
Stefan Zweig photo

Related topics