“Time and space are finite in extent, but they don't have any boundary or edge. They would be like the surface of the earth, but with two more dimensions.”

Source: Black Holes and Baby Universes

Last update Oct. 1, 2023. History

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Stephen Hawking 122
British theoretical physicist, cosmologist, and author 1942–2018

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“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property.”

Hans Reichenbach (1891–1953) American philosopher

The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes. ...but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.
The Philosophy of Space and Time (1928, tr. 1957)

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“That portion of the earth's surface which is owned and inhabited by the people of the United States is well adapted to be the home of one national family, and it is not well adapted for two or more. Its vast extent and its variety of climate and productions are of advantage in this age for”

Abraham Lincoln (1809–1865) 16th President of the United States

Source: 1860s, Second State of the Union address (1862)
Context: That portion of the earth's surface which is owned and inhabited by the people of the United States is well adapted to be the home of one national family, and it is not well adapted for two or more. Its vast extent and its variety of climate and productions are of advantage in this age for one people, whatever they might have been in former ages. Steam, telegraphs, and intelligence have brought these to be an advantageous combination for one united people.

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“I. Thesis. Finite elements of Space and Time. Antithesis.”

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“If the bee disappears from the surface of the earth, man would have no more than four years to live. No more bees, no more pollination … no more men!”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

A variant — "Professor Einstein, the learned scientist, once calculated that if all bees disappeared off the earth, four years later all humans would also have disappeared" — appears in The Irish Beekeeper, v.19-20, 1965-66, p74, citing Abeilles et Fleurs (Bees and Flowers, the house magazine of Union Nationale de l'Apiculture Française) for June 1965. Snopes.com mentions its use in a beekeepers' protest in 1994 in Europe http://www.snopes.com/quotes/einstein/bees.asp suggesting invention and attribution to Einstein for political reasons.
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“To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful… A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere… The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside… But… a being… unable to leave the surface… could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. …On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. …The spaces of zero and negative curvature are infinite, that of positive curvature is finite. …the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would… differ… by an amount too small to be appreciable… then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension…. our case with reference to three-dimensional space is exactly similar. …we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.”

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“A new consciousness is developing which sees the earth as a single organism and recognizes that an organism at war with itself is doomed. We are one planet. One of the great revelations of the age of space exploration is the image of the earth finite and lonely, somehow vulnerable, bearing the entire human species through the oceans of space and time.”

Carl Sagan (1934–1996) American astrophysicist, cosmologist, author and science educator

22 min 35 sec
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Context: Our global civilisation is clearly on the edge of failure and the most important task it faces, preserving the lives and well-being of its citizens and the future habitability of the planet. But if we're willing to live with the growing likelihood of nuclear war shouldn't we also been willing to explore vigorously every possible means to prevent nuclear war. Shouldn't we consider in every nation major changes in the traditional ways of doing things, a fundamental restructuring of economic political social and religious institutions. We've reached a point where there can be no more special interests or special cases, nuclear arms threaten every person on the Earth. Fundamental changes in society are sometimes labelled impractical or contrary to human nature, as if nuclear war were practical or as if there's only one human nature. But fundamental changes can clearly be made, we're surrounded by them. In the last two centuries abject slavery which was with us for thousands of years has almost entirely been eliminated in a stirring worldwide revolution. Women, systematically mistreated for millennia are gradually gaining the political and economic power traditionally denied them and some wars of aggression have recently been stopped or curtailed because of a revulsion felt by the people in the aggressor nations. The old appeals to racial, sexual, and religious chauvinism and to rabid nationalist fervor are beginning not to work. A new consciousness is developing which sees the earth as a single organism and recognizes that an organism at war with itself is doomed. We are one planet. One of the great revelations of the age of space exploration is the image of the earth finite and lonely, somehow vulnerable, bearing the entire human species through the oceans of space and time.

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