“I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect... Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.”

As quoted in Solid Shape (1990) by Jan J. Koenderink

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Carl Friedrich Gauss 50
German mathematician and physical scientist 1777–1855

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“Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.”

In the 1661 translation by Thomas Salusbury: … such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the neces­sity thereof, than which there can be no greater certainty." p. 92 (from the Archimedes Project http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?page=92;dir=galil_syste_065_en_1661;step=textonly)
In the original Italian: … tali sono le scienze matematiche pure, cioè la geometria e l’aritmetica, delle quali l’intelletto divino ne sa bene infinite proposizioni di piú, perché le sa tutte, ma di quelle poche intese dall’intelletto umano credo che la cognizione agguagli la divina nella certezza obiettiva, poiché arriva a comprenderne la necessità, sopra la quale non par che possa esser sicurezza maggiore." (from the copy at the Italian Wikisource).
Dialogue Concerning the Two Chief World Systems (1632)

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“Music is the arithmetic of sounds as optics is the geometry of light.”

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“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

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“Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”

Bernhard Riemann (1826–1866) German mathematician

Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein https://archive.org/details/TheEvolutionOfScientificThought (1927) p. 55.

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“Our civil rights have no dependence upon our religious opinions more than our opinions in physics or geometry.”

Thomas Jefferson (1743–1826) 3rd President of the United States of America

A Bill for Establishing Religious Freedom, Chapter 82 (1779). Published in The Works of Thomas Jefferson in Twelve Volumes http://oll.libertyfund.org/ToC/0054.php, Federal Edition, Paul Leicester Ford, ed., New York: G. P. Putnam's Sons, 1904, Vol. 1 http://oll.libertyfund.org/Texts/Jefferson0136/Works/0054-01_Bk.pdf, pp. 438–441. Comparison of Jefferson's proposed draft and the bill enacted http://web.archive.org/web/19990128135214/http://www.geocities.com/Athens/7842/bill-act.htm
1770s
Variant: Our civil rights have no dependence on our religious opinions any more than our opinions in physics or geometry...
Source: The Statute Of Virginia For Religious Freedom
Context: Well aware that the opinions and belief of men depend not on their own will, but follow involuntarily the evidence proposed to their minds; that Almighty God hath created the mind free, and manifested his supreme will that free it shall remain by making it altogether insusceptible of restraint; that all attempts to influence it by temporal punishments, or burthens, or by civil incapacitations, tend only to beget habits of hypocrisy and meanness, and are a departure from the plan of the holy author of our religion, who being lord both of body and mind, yet choose not to propagate it by coercions on either, as was in his Almighty power to do, but to exalt it by its influence on reason alone; that the impious presumption of legislature and ruler, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men, have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavoring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time: That to compel a man to furnish contributions of money for the propagation of opinions which he disbelieves and abhors, is sinful and tyrannical; … that our civil rights have no dependence on our religious opinions, any more than our opinions in physics or geometry; and therefore the proscribing any citizen as unworthy the public confidence by laying upon him an incapacity of being called to offices of trust or emolument, unless he profess or renounce this or that religions opinion, is depriving him injudiciously of those privileges and advantages to which, in common with his fellow-citizens, he has a natural right; that it tends also to corrupt the principles of that very religion it is meant to encourage, by bribing with a monopoly of worldly honours and emolumerits, those who will externally profess and conform to it; that though indeed these are criminals who do not withstand such temptation, yet neither are those innocent who lay the bait in their way; that the opinions of men are not the object of civil government, nor under its jurisdiction; that to suffer the civil magistrate to intrude his powers into the field of opinion and to restrain the profession or propagation of principles on supposition of their ill tendency is a dangerous fallacy, which at once destroys all religious liberty, … and finally, that truth is great and will prevail if left to herself; that she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict unless by human interposition disarmed of her natural weapons, free argument and debate; errors ceasing to be dangerous when it is permitted freely to contradict them.

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