“I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."”

— Benoît Mandelbrot

As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

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Benoît Mandelbrot 56

Polish-born, French and American mathematician 1924–2010

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“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.

“The pages of Euclid are like the opening bars of the music of the Grand Opera of Nature's great drama. So to say, they lift the veil and show to our vision a glimpse of a vast world of natural knowledge awaiting study.”

C. V. Raman (1888–1970) Indian physicist

Chandrasekhara Venkata Raman:A Legend of Modern Indian Science, 22 November 2013, Official Government of India's website Vigyan Prasar http://www.vigyanprasar.gov.in/scientists/cvraman/raman1.htm,

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

John D. Barrow (1952–2020) British scientist

The Artful Universe (1995)
Context: 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

“The term "paradigm," from the Greek paradeigma ("pattern"), was used by Kuhn to denote a conceptual framework shared by a community of scientists and providing them with model problems and solutions”

Fritjof Capra (1939) American physicist

Source: Uncommon Wisdom (1988), p. 22.

“The pattern of sex differences found in our species mirrors that found in most mammals and in many other animals. As such, considerations of parsimony suggest that the best explanation for the human differences will invoke evolutionary forces common to many species, rather than social forces unique to our own. When we find the standard pattern of differences in other, less culture-bound creatures, we inevitably explain this in evolutionary terms. It seems highly dubious, when we find exactly the same pattern in human beings, to say that, in the case of this one primate species, we must explain it in terms of an entirely different set of causes — learning or cumulative culture — which coincidentally replicates the pattern found throughout the rest of the animal kingdom. Anyone who wishes to adopt this position has a formidable task in front of them. They must explain why, in the hominin lineage uniquely, the standard evolved psychological differences suddenly became maladaptive, and thus why natural selection “wiped the slate clean” of any biological contribution to these differences. They must explain why natural selection eliminated the psychological differences but left the correlated physical differences intact. And they must explain why natural selection would eliminate the psychological differences and leave it all to learning, when learning simply replicated the same sex differences anyway. How could natural selection favor extreme flexibility with respect to sex differences if that flexibility was never exercised and was therefore invisible to selection?”

Steve Stewart-Williams (1971)

Source: The Ape that Thought It Was a Peacock: Does Evolutionary Psychology Exaggerate Human Sex Differences? (2013), pp. 142-143

“The efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute "a royal road to geometry"; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a schoolbook for so long.”

Thomas Little Heath (1861–1940) British civil servant and academic

The Thirteen Books of Euclid's Elements (1908)

“It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 30 Reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.

“Euclid's work will live long after all the text books of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid, it is necessary to know his language, and, so far as it can be traced, the history of the "elements" which he collected in his immortal work.”

Thomas Little Heath (1861–1940) British civil servant and academic

The Thirteen Books of Euclid's Elements (1908)

“J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 285; Cited in: Moritz (1914, 148); Persons and anecdotes

“It seems reasonable to suppose that conflict does exhibit many general patterns, that the patterns of conflict in industrial relations, international relations, interpersonal relations, and even animal life are not wholly different from one another, and that it is, therefore, worth looking for the common element. On the other hand, we should be surprised if there were no differences; the pattern of conflict in international relations, for instance, is not the same as in industrial or interpersonal relations. Just as it is important to perceive the similarities in different situations, so it is important to perceive the differences. These differences cannot be perceived, however, without a general theory to serve as a standard of comparison.”

Kenneth E. Boulding (1910–1993) British-American economist

Source: 1960s, Conflict and defense: A general theory, 1962, p. 2, partly cited in: Dennis Sandole (1998) A Comprehensive Mapping Of Conflict And Conflict. Resolution: A Three Pillar Approach http://www.gmu.edu/programs/icar/pcs/sandole.htm

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Benoît Mandelbrot 56

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