“Since the science of nature is conversant with magnitudes, motion, and time, each of which must necessarily be either infinite or finite…[we] should speculate the infinite, and consider whether it is or not; and if it is what it is. …[A]ll those who appear to have touched on a philosophy of this kind… consider it as a certain principle of beings. Some, indeed, as the Pythagoreans and Plato, consider it, per se, not as being an accident to any thing else, but as having an essential subsistence… the Pythagoreans… consider the infinite as subsisting in sensibles; for they do not make number to be separate; and they assert that what is beyond the heavens is infinite; but Plato says that beyond the heavens there is not any body, nor ideas, because these are no where: he affirms, however, that the infinite is both in sensibles, and in ideas. …Plato establishes two infinities, viz. the great and the small.”

— Aristotle , book Physics

Book III, Ch. IV, pp. 150-152.
Physics

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Aristotle 230

Classical Greek philosopher, student of Plato and founder o… -384–-321 BC

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“[B]ecause that which is finite is always bounded with reference to something… it is necessary that there should be no end… [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? …If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. …It is …especially the province of a natural philosopher to consider if there be a sensible infinite magnitude.”

Aristotle book Physics

Book III, Ch. V, p. 156.
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“All those… who discourse concerning nature, always subject a certain other nature of… elements, to the infinite… But no one of those who make the elements to be finite introduces infinity. Such, however, as make infinite elements, as Anaxagoras and Democritus, say that the infinite is continuous by contact. …Rationally, too, do all philosophers consider the infinite as a principle; for it cannot be in vain, nor can any other power be present with it than that of a principle: for all things are either the principle, or from the principle; but of the infinite there is no principle, since otherwise it would have an end. …it is also unbegotten and uncorruptible, as being a certain principle: for… end is the corruption of everything. …It likewise appears to comprehend and govern all things, as those assert who do not introduce other causes beside the infinite… It would seem also that this is divine: for it is immortal and indestructible, as Anaximander says, and most of the physiologists.”

Aristotle book Physics

Book III, Ch. IV, pp. 152-155.
Physics

“Since… nature is a principle of motion and mutation… it is necessary that we should not be ignorant of what motion is… But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. …[F]requently …those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous.”

Aristotle book Physics

Book III, Ch. I, pp. 135-136.
Physics

“Among these Ways, or any other whatever, of generating Magnitudes, the Primary and Chief is, that perform'd by local Motion, which all of them must in some Sort suppose, because without Motion, nothing can be generated or produced and therefore this must first be considered. The following Axiom of Aristotle concerning Motion is famous… He that is ignorant of Motion, must necessarily know nothing of Nature… in Nature every Thing created is produc'd by Motion, or certainly not without Motion.”

Isaac Barrow (1630–1677) English Christian theologian, and mathematician

p, 125
Geometrical Lectures (1735)

“[A]ll things as subsist from nature appear to contain in themselves a principle of motion and permanency; some according to place, others according to increase and diminuation; and others according to change in quality.”

Aristotle book Physics

Book II, Ch. I, p. 88.
Physics

“There is… something which is in energy only; and there is something which is both in energy and capacity. …of relatives, one is predicated as according to excess and defect: another according to the effective and passive, and, in short, the motive, and that which may be moved… Motion, however, has not a substance separate from things… But each of the categories subsists in a twofold manner in all things. Thus… one thing pertaining to it is form, and another privation. …So the species of motion and mutation are as many as those of being. But since in every genus of things, there is that which is in entelecheia, and that which is in capacity; motion is the entelecheia of that which is in capacity… That there is energy, therefore, and that a thing then happens to be moved, when this energy exists, and neither prior nor posterior to it, is manifest. … [N]either motion nor mutation can be placed in any other genus; nor have those who have advanced a different opinion concerning it spoken rightly. …for by some motion is said to be difference, inequality, and non-being; though it is not necessary that any of these should be moved… Neither is mutation into these, nor from these, rather than from their opposites. …The cause, however, why motion appears to be indefinite, is because it can neither be simply referred to the capacity, nor to the energy of beings. …[I]t is difficult to apprehend what motion is: for it is necessary to refer it either to privation, or to capacity, or to simple energy; but it does not appear that it can be any of these. The above-mentioned mode, therfore remains, viz. that it is a certain energy; but… difficult to be perceived, but which may have a subsistence.”

Aristotle book Physics

Book III, Ch. I, pp. 137-147.
Physics

“The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. We have considered it only in machines… [for which] the phenomenon is… incomplete. It becomes difficult to recognize its principles and study its laws. …[T]he principle of the production of motion by heat… must be considered independently of any mechanism or… particular agent. It is necessary to establish principles applicable not only to steam-engines but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated.”

Nicolas Léonard Sadi Carnot (1796–1832) French physicist, the "father of thermodynamics" (1796–1832)

Reflections on the Motive Power of Heat (1824)

“The true value of the Christian religion rests, not upon speculative views of the Creator, which must necessarily be different in each individual, according to the extent of the knowledge of the finite being, who employs his own feeble powers in contemplating the infinite : but it rests upon those doctrines of kindness and benevolence which that religion claims and enforces, not merely in favour of man himself but of every creature susceptible of pain or of happiness.”

Charles Babbage Passages from the life of a philosopher

"Passages from the life of a philosopher", The Basis Of Virtue Is Truth, p. 404-405
Passages from the Life of a Philosopher (1864)

“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 (1845–1918) mathematician, inventor of set theory

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

“Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 177

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