Friedrich Nietzsche quote: “Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature n…”

“If generation were in a straight line only, and there were no compensation or circle in nature, no turn or return into one another, then you know that all things would at last have the same form and pass into the same state, and there would be no more generation of them.”

Socrates (-470–-399 BC) classical Greek Athenian philosopher

Plato, Phaedo

“It is by the straight line and the circle that the first and most simple example and representation of all things may be demonstrated, whether such things be either non-existent or merely hidden under Nature's veils.”

John Dee (1527–1608) English mathematican, astrologer and antiquary

Theorem I
Monas Hieroglyphica (1564)

“A circle is a round straight line with a hole in the middle.”

Mark Twain (1835–1910) American author and humorist

Quoting a schoolchild in "English as She Is Taught"

“Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great? ], and inversely, when divided by any number he begets the infinitely great [small? ]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!”

Paul Carus (1852–1919) American philosopher

"Logical and Mathematical Thought?" in The Monist, Vol. 20 (1909-1910), p. 69

“Yes it was 1949. How I came to that. That's like how one gets to know a human being. It so happens that I've always had a preference – as everyone has prejudices and preferences – for the square as a shape in preference to the circle as a shape. And I have known for a long time that a circle always fools me by not telling me whether it's standing still or not. And if a circle circulates you don't see it. The outer curve looks the same whether it moves or does not move. So the square is much more honest and tells me that it is sitting on one line of the four, usually a horizontal one, as a basis. And I have also come to the conclusion that the square is a human invention, which makes it sympathetic to me. Because you don't see it in nature. As we do not see squares in nature, I thought that it is man-made. But I have corrected myself. Because squares exist in salt crystals, our daily salt. We know this because we can see it in the microscope. On the other hand, we believe we see circles in nature. But rarely precise ones. Mature, it seems, is not a mathematician. Probably there are no straight lines either. Particularly not since Einstein says in his theory of relativity that there is no straight line, rod knows whether there are or not, I don't. I still like to believe that the square is a human invention. And that tickles me. So when I have a preference for it then I can only say excuse me.”

Josef Albers (1888–1976) German-American artist and educator

Homage to the square' (1964), Oral history interview with Josef Albers' (1968)

“I did my best to go in a circle, hoping in this way to go in a straight line.”

Samuel Beckett book Molloy

Molloy (1951)
Context: Having heard, or more probably read somewhere, in the days when I thought I would be well advised to educate myself, or amuse myself, or stupefy myself, or kill time, that when a man in a forest thinks he is going forward in a straight line, in reality he is going in a circle, I did my best to go in a circle, hoping in this way to go in a straight line. For I stopped being half-witted and became sly, whenever I took the trouble … and if I did not go in a rigorously straight line, with my system of going in a circle, at least I did not go in a circle, and that was something.

“But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.”

Roger Joseph Boscovich (1711–1787) Croat-Italian physicist

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
"Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham.
"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

“She was strikingly ignorant, but he had known that from the start. What he had not known was how quickly her ignorance would cease to seem charming and would begin to seem contemptible.”

Robert Silverberg book Thorns

Source: Thorns (1967), Chapter 21, “And Southward Aye We Fled” (p. 106)

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little. In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.”

Immanuel Kant (1724–1804) German philosopher

Preface, Tr. Bax (1883) citing Isaac Newton's Principia
(1786)

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little.”

Immanuel Kant book Metaphysical Foundations of Natural Science

In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.
Preface, Tr. Bax (1883) citing Isaac Newton's Principia
Metaphysical Foundations of Natural Science (1786)

“Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.”

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