„It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind.“

[On Riemannian manifolds of four dimensions, Bulletin of the American Mathematical Society, 51, 12, 1945, 964–971, http://www.ams.org/journals/bull/1945-51-12/S0002-9904-1945-08483-3/S0002-9904-1945-08483-3.pdf]

Adopted from Wikiquote. Last update June 3, 2021. History
Shiing-Shen Chern photo
Shiing-Shen Chern6
mathematician (1911–2004), born in China and later acquirin… 1911 - 2004

Related quotes

Roger Shepard photo
Marcel Duchamp photo
Benjamin Creme photo
Willem de Sitter photo
Johan Huizinga photo
François Englert photo
Jomo Kenyatta photo

„The Bible is not the property of one nation or of one group of people,“

—  Jomo Kenyatta First prime minister and first president of Kenya 1893 - 1978

Context: Whether I am a Christian or not is none of your business. Mr. Speaker, I have nothing to add. My friend, I can see that your philosophy is running short; The Bible is not the property of one nation or of one group of people, it can be quoted by anyone, even you. I have nothing further to add to the answer that I have already given. I do, however, call upon the Kenya nation to wake up and help itself. Thank you.

Steven Pinker photo

„Equality is not the empirical claim that all groups of humans are interchangeable; it is the moral principle that individuals should not be judged or constrained by the average properties of their group.“

—  Steven Pinker, book The Blank Slate

p. 485 http://books.google.com/books?id=ePNi4ZqYdVQC&q=%22humans+are+interchangeable%22
The Blank Slate (2002)
Source: The Blank Slate: The Modern Denial of Human Nature
Context: [E]quality is not the empirical claim that all groups of humans are interchangeable; it is the moral principle that individuals should not be judged or constrained by the average properties of their group. … If we recognize this principle, no one has to spin myths about the indistinguishability of the sexes to justify equality.

Hans Reichenbach photo

„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 American philosopher 1891 - 1953

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)

Friedrich Hayek photo
Charles Darwin photo
Reggie Fils-Aimé photo
Samuel P. Huntington photo

„The most important groupings of states are no longer the three blocs of the Cold War but rather the world’s seven or eight major civilizations.“

—  Samuel P. Huntington American political scientist 1927 - 2008

Source: The Clash of Civilizations and the Remaking of World Order (1996), Ch. 1: The New Era in World Politics, § 2 : A Multipolar, Multicivilizational World
Context: In the post-Cold War world, for the first time in history, global politics has become multipolar and multicivilizational. During most of human existence, contacts between civilizations were intermittent or nonexistent. Then, with the beginning of the modern era, about A. D. 1500, global politics assumed two dimensions. For over four hundred years, the nation states of the West — Britain, France, Spain, Austria, Prussia, Germany, the United States, and others — constituted a multipolar international system within Western civilization and interacted, competed, and fought wars with each other. At the same time, Western nations also expanded, conquered, colonized, or decisively influenced every other civilization. During the Cold War global politics became bipolar and the world was divided into three parts. A group of mostly wealthy and democratic societies, led by the United States, was engaged in a pervasive ideological, political, economic, and, at times, military competition with a group of somewhat poorer communist societies associated with and led by the Soviet Union. Much of this conflict occurred in the Third World outside these two camps, composed of countries which often were poor, lacked political stability, were recently independent, and claimed to be nonaligned.
In the late 1980s the communist world collapsed, and the Cold War international system became history. In the post-Cold War world, the most important distinctions among peoples are not ideological, political, or economic. They are cultural. Peoples and nations are attempting to answer the most basic question humans can face: Who are we? And they are answering that question in the traditional way human beings have answered it, by reference to the things that mean most to them. People define themselves in terms of ancestry, religion, language, history, values, customs, and institutions. They identify with cultural groups: tribes, ethnic groups, religious communities, nations, and, at the broadest level, civilizations. People use politics not just to advance their interests but also to define their identity. We know who we are only when we know who we are not and often only when we know whom we are against.
Nation states remain the principal actors in world affairs. Their behavior is shaped as in the past by the pursuit of power and wealth, but it is also shaped by cultural preferences, commonalities, and differences. The most important groupings of states are no longer the three blocs of the Cold War but rather the world’s seven or eight major civilizations. Non-Western societies, particularly in East Asia, are developing their economic wealth and creating the basis for enhanced military power and political influence. As their power and self-confidence increase, non-Western societies increasingly assert their own cultural values and reject those “imposed” on them by the West.

Hermann Weyl photo

„Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself.“

—  Hermann Weyl German mathematician 1885 - 1955

On the foundations of general infinitesimal geometry. Bull. Amer. Math. Soc. 35 (1929) 716–725 [10.1090/S0002-9904-1929-04812-2] (quote on p. 716)

Henri Poincaré photo
Hans Reichenbach photo
Richard Nixon photo

„Nixon: Within groups, there are geniuses. There are geniuses within black groups. There are more within Asian groups … This is knowledge that is better not to know.“

—  Richard Nixon 37th President of the United States of America 1913 - 1994

Fall of 1971, conversation with Harvard professor Daniel Patrick Moynihan http://nixontapeaudio.org/chron2/rmn_e010b.mp3; as qtd. in Tim Naftali, “Ronald Reagan’s Long-Hidden Racist Conversation With Richard Nixon” https://www.theatlantic.com/ideas/archive/2019/07/ronald-reagans-racist-conversation-richard-nixon/595102/, The Atlantic, (Jul 30, 2019)
1970s, Tape transcripts (1971)

E. W. Hobson photo

Related topics