On Numbers: The Infinity in Numbers

“When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]” (Aristotle)

“But of all other ideas, it is number, which I think furnishes us with the clearest and most distinct idea of infinity we are capable of.” (John Locke)

"For any number there exists a corresponding even number which is its double. Hence the number of all numbers is not greater than the number of even numbers, that is, the whole is not greater than the part." (Gottfried W Leibniz)

“I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding […]” (Richard Dedekind, “On Continuity and Irrational Numbers”, 1872)

 “If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.” (Bertrand Russell)

"The prototype of all infinite processes is repetition. […] Our very concept of the infinite derives from the notion that what has been said or done once can always be repeated.” (Tobias Dantzig, “Number: The Language of Science”, 1930)

“The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]” (Hermann Weyl, “Mathematics and Logic”, 1946)

"[…] infinity is not a large number or any kind of number at all; at least of the sort we think of when we say 'number'. It certainly isn't the largest number that could exist, for there isn't any such thing." (Isaac Asimov)

"Each act of creation could be symbolized as a particular product of infinity and zero. From each such product could emerge a particular entity of which the appropriate symbol was a particular number." (Srinivasa Ramanujan)

“Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite.” (Paul Erdős)