“The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction.”

100 Years of Mathematics: a Personal Viewpoint (1981)

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George Frederick James Temple21

British mathematician 1901–1992

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“Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).”

Henry Burchard Fine (1858–1928) American academic

Source: The Number-System of Algebra, (1890), p. 86; Reported in Moritz (1914, 282)

“Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

“You can ask the question about these ancient topics, such as perfect numbers and amicable numbers… and ask, are these good problems… I'd like to give a small amount of evidence… that they are… [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra..”

Carl Pomerance (1944) American mathematician

"Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory" https://www.youtube.com/watch?v=5cU0g9dI1S8&t=9m40s (July, 2013) Erdős Centennial Conference, Budapest.

“The Hindus saw clearly that if the arithmetic operations… were properly defined for negative numbers, these numbers could be employed to as good advantage as people had previously derived from positive numbers. …To people to whom the word number had always meant positive whole numbers and positive fractions, the very idea that there could be other numbers came hard. For many centuries negative numbers were either rejected or treated as second-class citizens.
What was especially difficult for mathematicians to swallow was that negative numbers could be acceptable roots of equations.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 51.

“Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity there is no such jump, and because jump is missing in the world of quantity it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.”

Gregory Bateson (1904–1980) English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist

Bateson (1978) " Number is Different from Quantity http://www.oikos.org/batesnumber.htm". In: CoEvolution Quarterly, Spring 1978, pp. 44-46

“As for negative numbers… most mathematicians of the sixteenth and seventeenth centuries did not accept them… In the fifteenth century Nicolas Chuquet and, in the sixteenth, Stifel both spoke of negative numbers as absurd numbers. …Descartes accepted them, in part. …he had shown that, given an equation, one can obtain another whose roots are larger than the original one by any given quantity. Thus an equation with negative roots could be transformed into one with positive roots. Since we can turn false roots into real roots, Descartes was willing to accept negative numbers. Pascal regarded the subtraction of 4 from zero as utter nonsense.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 252.

“1. Zero is a number.
2. The successor of any number is another number.
3. There are no two numbers with the same successor.
4. Zero is not the successor of a number.
5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers.”

Giuseppe Peano (1858–1932) Italian mathematician

As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
Peano axioms

“The physicist … engages in complex and difficult calculations, involving the manipulating of ideal, mathematical quantities that, at first glance, are wholly lacking in the music of the living world and the beauty of the resplendent cosmos. It would seem as if there exists no relationship between these quantities and reality. Yet these ideal numbers that cannot be grasped by one's senses, these numbers that only are meaningful from within the system itself, only meaningful as part of abstract mathematical functions, symbolize the image of existence.”

Joseph B. Soloveitchik (1903–1993) American theologian

Source: Halakhic Man (1983), p. 83

“During the renaissances in thinking led by Shakespeare and Goethe, the first new development in memory techniques for 1,700 years appeared: the Major System. This was the first system that enabled the user to transfer easily and instantaneously from numbers to letters, thus creating the opportunity for a system that could stretch from zero to an infinite number, and which allowed the user to translate any word into its own special number, and any number into its own special letter. This multiplied the opportunity for developing memory techniques 100-fold.”

Tony Buzan (1942–2019) British psychologist

The Mind Map Book, Buzan and Buzan (1991)

“Number is different from quantity.”

Gregory Bateson (1904–1980) English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist

Source: Mind and Nature, a necessary unity, 1988, p. 118

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George Frederick James Temple21

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