“In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.”

— Georg Cantor

"Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)

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Georg Cantor 27

mathematician, inventor of set theory 1845–1918

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“There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

“In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead
Context: This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers... Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
... The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
... to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem...
In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.<!--pp.10-12

“From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“Before we can proceed to a formal definition of conflict we must examine another concept, that of behavior space. The position of a behavior unit at a moment of time is defined by a set of values (subset, to be technical) of a set of variables that defines the behavior unit. These variables need not be continuous or quantitatively measurable. The different values of a variable must, however, be capable of simple ordering; that is, of any two values it must be possible to say that one is 'after' (higher, lefter, brighter than) the other.”

Kenneth E. Boulding (1910–1993) British-American economist

Peace Science Society (International) (1975) Papers - Volumes 24-29. p. 53 summarized: "Boulding begins by explaining what he believes are the four basic concepts to describe a conflict in an analytical way : (1) the party; (2) the behavior space; (3) competition; (4) conflict."
Source: 1960s, Conflict and defense: A general theory, 1962, p. 3

“Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)

“To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.”

Leonhard Euler (1707–1783) Swiss mathematician

As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214

“In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.”

Hermann Weyl Invariants

Weyl, Hermann. Invariants. Duke Math. J. 5 (1939), no. 3, 489--502. doi:10.1215/S0012-7094-39-00540-5. http://projecteuclid.org/euclid.dmj/1077491405.

“What makes a piece of mathematical economics not only mathematics but also economics is, I believe, this: When we set up a system of theoretical relationships and use economic names for the otherwise purely theoretical variables involved, we have in mind some actual experiment, or some design of an experiment, which we could at least imagine arranging, in order to measure those quantities in real economic life that we think might obey the laws imposed on their theoretical namesakes.”

Trygve Haavelmo (1911–1999) Norwegian economist and econometrician

Trygve Haavelmo, "The probability approach in econometrics" in: Supplement to Econometrica. 12 91944), p. 5; Cited in Pearl (2012, 1-2)

“Public policy does not admit of definition and is not easily explained. It is a variable quantity; it must vary and does vary with the habits, capacities, and opportunities of the public.”

Arthur Kekewich (1832–1907) British judge

Davies v. Davies (1887), L. R. 36 C. D. 364; see also Egerton v. Earl Brownlow, 4 H. L. C. 1.

“The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

"Mitteilungen" (1887-8)

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Georg Cantor 27

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