„A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid.“

—  Freeman Dyson, buch Infinite in All Directions

Quelle: Infinite in All Directions (1988), Ch. 2 : Butterflies and Superstrings, p. 17
Kontext: Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry?... It cannot be answered by a definition.

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Freeman Dyson Foto
Freeman Dyson1
englisch-US-amerikanischer Physiker und Mathematiker 1923

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Benoît Mandelbrot Foto
Freeman Dyson Foto

„The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? …It cannot be answered by a definition.“

—  Freeman Dyson, buch Infinite in All Directions

Quelle: Infinite in All Directions (1988), Ch. 2 : Butterflies and Superstrings, p. 17
Kontext: Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry?... It cannot be answered by a definition.

Robert Anton Wilson Foto
Buckminster Fuller Foto

„A pattern has an integrity independent of the medium by virtue of which you have received the information that it exists. Each of the chemical elements is a pattern integrity. Each individual is a pattern integrity. The pattern integrity of the human individual is evolutionary and not static.“

—  Buckminster Fuller American architect, systems theorist, author, designer, inventor and futurist 1895 - 1983

Pattern Integrity 505.201 http://www.rwgrayprojects.com/synergetics/s05/p0400.html#505
1970s, Synergetics: Explorations in the Geometry of Thinking (1975), "Synergy" onwards

José Saramago Foto
Nicomachus Foto

„If geometry exists, arithmetic must also needs be implied“

—  Nicomachus Ancient Greek mathematician 60 - 120

Nicomachus of Gerasa: Introduction to Arithmetic (1926)
Kontext: If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.<!--Book I, Chapter IV

E. W. Hobson Foto
Antoinette Brown Blackwell Foto

„The antagonism is only that of action and reaction, which are but two phases of the same process—opposing phases which exist everywhere, and which must exist, or action itself cease, and death reign universally.“

—  Antoinette Brown Blackwell American minister 1825 - 1921

September 1874, Popular Science Monthly Vol. 5, Article: The Alleged Antagonism Between Growth and Reproduction , p. 607
The Alleged Antagonism Between Growth and Reproduction (1874)

Benoît Mandelbrot Foto

„There is no single rule that governs the use of geometry. I don't think that one exists.“

—  Benoît Mandelbrot Polish-born, French and American mathematician 1924 - 2010

New Scientist interview (2004)

John D. Barrow Foto

„The laws of Nature are based upon the existence of a pattern,“

—  John D. Barrow British scientist 1952

The Artful Universe (1995)
Kontext: The laws of Nature are based upon the existence of a pattern, linking one state of affairs to another; and where there is pattern, there is symmetry. Yet... the symmetries that the laws enshrine are broken in... outcomes. Suppose that we balance a needle on its point and then release it. The law of gravity, which governs its subsequent motion, is perfectly democratic. It has no preference for any particular direction in the Universe: it is symmetrical in this respect. Yet, when the needle falls, it must fall in a particular direction. The directional symmetry of the underlying law is broken, therefore... By the same token, the fallen needle hides the symmetry of the law... Such 'symmetry-breaking' governs much of what we see in the Universe... It allows a Universe governed by a small number of symmetrical laws to manifest an infinite diversity of complex, asymmetrical states. This is how the Universe can be at once, simple and complicated.<!-- Ch. 2, pp. 36-37

Russell L. Ackoff Foto
John Updike Foto

„Existence itself does not feel horrible; it feels like an ecstasy, rather, which we have only to be still to experience.“

—  John Updike American novelist, poet, short story writer, art critic, and literary critic 1932 - 2009

Quelle: Self-Consciousness : Memoirs (1989), Ch. 6

„Parts and wholes evolve in consequence of their relationship, and the relationship itself evolves.“

—  Richard C. Lewontin American evolutionary biologist 1929

The Dialectical Biologist (1985), co-written with Richard Levins, Introduction, p. 3.
Kontext: Parts and wholes evolve in consequence of their relationship, and the relationship itself evolves. These are the properties of things that we call dialectical: that one thing cannot exist without the other, that one acquires its properties from its relation to the other, that the properties of both evolve as a consequence of their interpenetration.

Werner Heisenberg Foto
Maimónides Foto
Georg Wilhelm Friedrich Hegel Foto

„Spirit, on the contrary, may be defined as that which has its center in itself. It has not a unity outside itself, but has already found it; it exists in and with itself. Matter has its essence out of itself; Spirit is self-contained existence (Bei-sich-selbst-seyn). Now this is Freedom, exactly. For if I am dependent, my being is referred to something else which I am not; I cannot exist independently of something external. I am free, on the contrary, when my existence depends upon myself. This self-contained existence of Spirit is none other than self-consciousness consciousness of one's own being. Two things must be distinguished in consciousness; first, the fact that I know; secondly, what I know. In self-consciousness these are merged in one; for Spirit knows itself. It involves an appreciation of its own nature, as also an energy enabling it to realise itself; to make itself actually that which it is potentially.“

—  Georg Wilhelm Friedrich Hegel, buch Lectures on the Philosophy of History

Lectures on the History of History Vol 1 p. 18 John Sibree translation (1857), 1914
Lectures on the Philosophy of History (1832), Volume 1
Kontext: The nature of Spirit may be understood by a glance at its direct opposite Matter. As the essence of Matter is Gravity, so, on the other hand, we may affirm that the substance, the essence of Spirit is Freedom. All will readily assent to the doctrine that Spirit, among other properties, is also endowed with Freedom; but philosophy teaches that all the qualities of Spirit exist only through Freedom; that all are but means for attaining Freedom; that all seek and produce this and this alone. It is a result of speculative Philosophy, that Freedom is the sole truth of Spirit. Matter possesses gravity in virtue of its tendency towards a central point. It is essentially composite; consisting of parts that exclude each other. It seeks its Unity; and therefore exhibits itself as self- destructive, as verging towards its opposite [an indivisible point]. If it could attain this, it would be Matter no longer, it would have perished. It strives after the realization of its Idea; for in Unity it exists ideally. Spirit, on the contrary, may be defined as that which has its center in itself. It has not a unity outside itself, but has already found it; it exists in and with itself. Matter has its essence out of itself; Spirit is self-contained existence (Bei-sich-selbst-seyn). Now this is Freedom, exactly. For if I am dependent, my being is referred to something else which I am not; I cannot exist independently of something external. I am free, on the contrary, when my existence depends upon myself. This self-contained existence of Spirit is none other than self-consciousness consciousness of one's own being. Two things must be distinguished in consciousness; first, the fact that I know; secondly, what I know. In self-consciousness these are merged in one; for Spirit knows itself. It involves an appreciation of its own nature, as also an energy enabling it to realise itself; to make itself actually that which it is potentially.

Maimónides Foto

„That which is produced with intention has passed over from non-existence to existence.“

—  Maimónides, buch The Guide for the Perplexed

Quelle: Guide for the Perplexed (c. 1190), Part III, Ch.13

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