“In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i. e., in the following principle:
"If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions."
…every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed.”

— Richard Dedekind

Stetigkeit und irrationale Zahlen (1872)

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 "In the preceding section attention was called to the fact that every point p of the straight line produces a separation…" by Richard Dedekind?

Richard Dedekind 13

German mathematician 1831–1916

Proclus (412–485) Greek philosopher

Book III. Concerning Petitions and Axioms.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 2 (1789)

“A straight line is not the shortest distance between two points.”

Madeleine L'Engle (1918–2007) American writer

Source: A Wrinkle in Time: With Related Readings

“…the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous, although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i. e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. …such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“…the relation of betweenness on the torus is undetermined for curves that cannot be contracted to a point [e. g., circles around a doughnut hole], i. e., for three of such curves it is not uniquely determined which of them lies between the other two... This indeterminateness… has the consequence that such a curve [alone] does not divide the surface of the torus into two separate domains; between points to the "right" and to the "left" of the line.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“In philosophy, as in politics, the longest distance between two points is a straight line.”

Will Durant (1885–1981) American historian, philosopher and writer

Source: The Story of Philosophy: The Lives and Opinions of the World's Greatest Philosophers

“The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 9.
On the Equilibrium of Planes

“The straight line is regarded as the shortest distance between two people, as if they were points.”

Theodor W. Adorno book Minima Moralia

Nun gilt für die kürzeste Verbindung zwischen zwei Personen die Gerade, so als ob sie Punkte wären.
E. Jephcott, trans. (1974), § 20
Minima Moralia (1951)

“In a constantly revolving circle every point is simultaneously a point of departure and a point of return. If we interrupt the rotation, not every point of departure is a point of return.”

Karl Marx (1818–1883) German philosopher, economist, sociologist, journalist and revolutionary socialist

Vol. II, Ch. IV, p. 104.
(Buch II) (1893)

“Every point is a turning point.”

Pat Murphy book There and Back Again

Catch phrase, often repeated
There and Back Again (1999)

“In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.”

Archimedes book On the Equilibrium of Planes

Book 1, Proposition 13.
On the Equilibrium of Planes

Help us to complete the source, original and additional information

Richard Dedekind 13

Related quotes

Related topics