Infinite and Geometry

“The knowledge of which geometry aims is the knowledge of the eternal." (Plato)

"You say that just as space consists of an infinity of contiguous points, so time is but an infinite collection of contiguous instants? Good! Consider, then, an arrow in its flight. At any instant its extremity occupies a definite point in its path. Now, while occupying this position it must be at rest there. But how can a point be motionless and yet in motion at the same time?” (Zeno)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)

“Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish.” (Aristotle, Physics)

"A finite straight line can be extended indefinitely to make an infinitely long straight line." (Euclid’s postulate)

"Given a straight line and any point off to the side of it, there is, through that point, one and only one line that is parallel to the given line." (Euclid’s postulate)

"We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads." (Voltaire)

“Of late the speculations about Infinities have run so high, and grown to such strange notions, as have occasioned no small scruples and disputes among the geometers of the present age. Some there are of great note who, not contented with holding that finite lines may be divided into an infinite number of parts, do yet further maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These I say assert there are infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts.” (George Berkeley, “The Principles of Human Knowledge”, 1710)

“The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology.” (John Theodore Merz, “History of European Thought in the Nineteenth Century”, 1903)