Number

Pythagoras... assumed as first principles the numbers and symmetries existing among them, which he calls harmonies, and the elements compounded of both, that are called geometrical. ...he says that the nature of Number is the Decad.

— Number

With the intrusion of irrational numbers to disrupt the integral harmonies of the Pythagorean cosmos, a controversy that has raged off and on for well over two thousand years began: is the mathematical infinite a safe concept in mathematical reasoning, safe in the sense that contradictions will not result from the use of this infinite, subject to certain prescribed conditions?

— Number

In Greek theoretical mathematics (as distinguished from practical or commercial arithmetic) a fraction that we would write as a/b was not regarded as a number, as a single entity, but as a relationship or a : b between the whole numbers a and b. Thus the ratio a : b was, in modern terms, simply an ordered pair, rather than a rational number.

— Number

In past centuries it was widely accepted that an understanding of, as well as a facility with numbers, is an essential part of an education. ...This book has been written with an intention of showing that numbers have been the centre of man's awareness of his surroundings since well before any times of which we have surviving records. It will show that numbers have provided an answer to man's cultural needs at least since any form of organized human society came into being.

— Number

It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since

— Number

Bourgeois society is ruled by equivalence. It makes dissimilar things comparable by reducing them to abstract quantities. For the Enlightenment, anything which cannot be resolved into numbers, and ultimately into one, is illusion; modern positivism consigns it to poetry.

— Number

[Number is] the commanding and self-begotten container of the eternal duration of mundane concerns.

— Number

[A]ll things which can be known have number; for it is not possible that without number anything can either be conceived or known.

— Number

All things, at least those we know, contain number; for it is evident that nothing whatever can either be thought or known, without number. Number has two distinct kinds: the odd, and the even, and a third, derived from a mingling of the other two kinds, the even-odd. Each of its subspecies is susceptible of many very numerous varieties; which each manifests individually.

— Number

Number is the ruler of forms and ideas, and the cause of gods and daemons.

— Number

Number rules the universe.

— Number

The evolution of number into the 'transfinite' was included only to emphasize the power of the forces acting within mathematics to compel this development—even against the philosophy of its most prominent creator, George Cantor (...numbers were extended, along with their arithmetic, to the non-finite, not as a mathematical whim, but for reasons of strong internal stresses.)

— Number

For Plato, the first upward steps out of the cave toward wisdom begin with mastery of the arts of number. This put thought on the path of representation and mathematical objectification. Mathematics' more concrete, everyday rule—to serve the needs of power—makes this path the history of oppression.

— Number