“That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices.”

— Georg Cantor

Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472

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Georg Cantor 27

mathematician, inventor of set theory 1845–1918

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“It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning.”

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“1. Zero is a number.
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“Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers.”

Giuseppe Peano (1858–1932) Italian mathematician

As expressed in Galileo's Finger: The Ten Great Ideas of Science (2003) by Peter Atkins, Ch. 10 "Arithmetic : The Limits of Reason", p. 333
Peano axioms
Context: 1. 0 is a number.
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“When we attempt to conceive the number of particles in an atmosphere, it is somewhat like attempting to conceive the number of stars in the universe; we are confounded with the thought. But if we limit the subject, by taking a given volume of any gas, we seem persuaded that, let the divisions be ever so minute, the number of particles must be finite; just as in a given space of the universe, the number of stars and planets cannot be infinite.”

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Vol. III, p. 473 - I have read this page twice and cannot find this quote.
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“What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.”

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As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178

“A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis.”

Emmy Noether (1882–1935) German mathematician

As quoted in Morris Kline, Mathematical Thought From Ancient to Modern Times (1972) p. 1153.

“As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved.”

Diophantus Arithmetica

As quoted by James Gow, A Short History of Greek Mathematics https://books.google.com/books?id=9d8DAAAAMAAJ (1884)
Arithmetica (c. 250 AD)

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Georg Cantor 27

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