“Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found.”

— Galileo Galilei

Salviati, First Day, Stillman Drake translation (1974)
Dialogues and Mathematical Demonstrations Concerning Two New Sciences (1638)

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Galileo Galilei70

Italian mathematician, physicist, philosopher and astronomer 1564–1642

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“What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers.”

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

From Kant to Hilbert (1996)

“What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.”

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

As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178

“For how shall the finite comprehend the infinite?”

Baruch Spinoza (1632–1677) Dutch philosopher

George Boole, An Investigation of the Laws of Thought (1854) Ch. 13. Clarke and Spinoza, pp. 216-217. https://books.google.com/books?id=DqwAAAAAcAAJ&pg=PA216 <br class="br">Context: It is not possible, I think, to rise from the perusal of the arguments of Clark and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely à priori, the existence of an Infinite Being, His attributes, and His relation to the universe. The fundamental principle of all such speculations, viz. that whatever we can clearly conceive, must exist, fails to accomplish its end, even when its truth is admitted. For how shall the finite comprehend the infinite? Yet must the possibility of such conception be granted, and in something more than the sense of a mere withdrawal of the limits of phænomal existence, before any solid ground can be established for the knowledge, à priori, of things infinite and eternal.

“[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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“A finite game is played for the purpose of winning, an infinite game for the purpose of continuing the play.”

James P. Carse American academic

Source: Finite and Infinite Games: A Vision of Life as Play and Possibility

“[I]t is impossible that each of the elements should be infinite. For that is body which has interval on all sides; and that is infinite which has extension without bound.”

Aristotle book Physics

Book III, Ch. VII, p. 159.
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“For man to be able to live he must either not see the infinite, or have such an explanation of the meaning of life as will connect the finite with the infinite.”

Leo Tolstoy (1828–1910) Russian writer

Source: Confession (1882), Ch. 9

“How good would it be if one could die by throwing oneself into an infinite void.”

Emil M. Cioran (1911–1995) Romanian philosopher and essayist

Source: On the Heights of Despair (1934)

“The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Mysticism and Logic and Other Essays http://archive.org/stream/mysticism00russuoft/mysticism00russuoft_djvu.txt (1918), Ch. 5: Mathematics and the Metaphysicians

“To romanticize the world is to make us aware of the magic, mystery and wonder of the world; it is to educate the senses to see the ordinary as extraordinary, the familiar as strange, the mundane as sacred, the finite as infinite.”

Novalis (1772–1801) German poet and writer

As quoted in "Bildung in Early German Romanticism" by Frederick C. Beiser, in Philosophers on Education : Historical Perspectives (1998) by Amélie Rorty, p. 294

“Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found.”

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Galileo Galilei70

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