“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 , book Infinite in All Directions

Source: Infinite in All Directions (1988), Ch. 2 : Butterflies and Superstrings, p. 17
Context: 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 90

theoretical physicist and mathematician 1923

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“Existence is larger than any model that is not itself the exact size of existence (which has no size…)”

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“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 book Infinite in All Directions

“A new point is determined in Euclidean Geometry exclusively in one of the three following ways:
Having given four points A, B, C, D, not all incident on the same straight line, then
(1) Whenever a point P exists which is incident both on (A, B) and on (C, D), that point is regarded as determinate.
(2) Whenever a point P exists which is incident both on the straight line (A, B) and on the circle C(D), that point is regarded as determinate.
(3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate.
The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules (1), (2) and (3). Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite.
In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to (1) the ruler is required, corresponding to (2) both ruler and compass, and corresponding to (3) the compass only.
…it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of (3) alone…”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 7-8

“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 (1895–1983) American architect, systems theorist, author, designer, inventor and futurist

Pattern Integrity 505.201 http://www.rwgrayprojects.com/synergetics/s05/p0400.html#505
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“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 (1825–1921) American minister

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The Alleged Antagonism Between Growth and Reproduction (1874)

“Each part in itself constitutes the whole to which it belongs.”

José Saramago book The Cave

Source: The Cave (2000), p. 68 (Vintage 2003)

“Relationships are all there is. Everything in the universe only exists because it is in relationship to everything else. Nothing exists in isolation. We have to stop pretending we are individuals that can go it alone.”

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Source: Turning to one another (2002), p. 19

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

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)
Context: 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

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

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Freeman Dyson 90

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