The Infinite and Its Difficulties II

“The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.” (Thomas Aquinas, “Summa theologia”, 13th century) 

"[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but this […] is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.” (Galileo Galilei, "Two New Sciences", 1638)

“Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness; Imagine what they are when combined. In spite of this men cannot refrain from discussing them.” (Galileo Galilei, "Two New Sciences", 1638)

“Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power.” (Thomas Hobbes, "Of Man", 1658)

“Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed.” (Blaise Pascal, "Pensées", 1670)

“Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

 “A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined.” (George B Mathews, “Theory of Numbers”, 1892)

“Like children who are not permitted to do certain things, we are not permitted by nature to think in terms of infinity.” (Robert Tuttle Morris, “Microbes and Men”, 1916)

 "The infinite in mathematics is always unruly unless it is properly treated."  (Edward Kasner & James Newman, “Mathematics and the Imagination”, 1940)

“I am incapable of conceiving infinity, and yet I do not accept finity.” (Simone de Beauvoir, “La Vieillesse”, 1970)