The philosophy of Aristotle, expressed in such works as his Physics, held that although space was finite, with only void existing beyond the outermost sphere of the heavens, time was infinite. This caused problems for mediaeval Islamic, Jewish, and Christian philosophers, who were unable to reconcile the Aristotelian conception of the eternal with the Abrahamic view of Creation.
In contrast to ancient Greek philosophers who believed that the universe had an infinite past with no beginning, medieval philosophers and theologians developed the concept of the universe having a finite past with a beginning. This view was inspired by the creation doctrine shared by the three Abrahamic religions: Judaism, Christianity and Islam.
Prior to Maimonides, it was held that it was possible to prove, philosophically, creation theory. The Kalam cosmological argument held that creation was provable, for example. Maimonides himself held that neither creation nor Aristotle's infinite time were provable, or at least that no proof was available. (According to scholars of his work, he didn't make a formal distinction between unprovability and the simple absence of proof.) Thomas Aquinas was influenced by this belief, and held in his Summa Theologica that neither hypothesis was demonstrable. Some of Maimonides' Jewish successors, including Gersonides and Crescas, conversely held that the question was decidable, philosophically.
John Philoponus was probably the first to use the argument that infinite time is impossible, establishing temporal finitism. He was followed by many others including Al-Kindi, Saadia Gaon, Al-Ghazali, St. Bonaventure and Immanuel Kant (in his First Antinomy). The argument was revisited once again by William Lane Craig in light of the idea of transfinite numbers in modern mathematics.
Philoponus' arguments for temporal finitism were severalfold. Contra Aristotlem has been lost, and is chiefly known through the citations used by Simplicius of Cilicia in his commentaries on Aristotle's Physics and De Caelo. Philoponus' refutation of Aristotle extended to six books, the first five addressing De Caelo and the sixth addressing Physics, and from comments on Philoponus made by Simplicius can be deduced to have been quite lengthy.
A full exposition of Philoponus' several arguments, as reported by Simplicius, can be found in Sorabji, listed in Further reading. One such argument was based upon Aristotle's own theorem that there were not multiple infinities, and ran as follows: If time were infinite, then as the universe continued in existence for another hour, the infinity of its age since creation at the end of that hour must be one hour greater than the infinity of its age since creation at the start of that hour. But since Aristotle holds that such treatments of infinity are impossible and ridiculous, the world cannot have existed for infinite time.
Philoponus' works were adopted by many, most notably; early Muslim philosopher, Al-Kindi (Alkindus); the Jewish philosopher, Saadia Gaon (Saadia ben Joseph); and the Muslim theologian, Al-Ghazali (Algazel). They used his two logical arguments against an infinite past, the first being the "argument from the impossibility of the existence of an actual infinite", which states:
- "An actual infinite cannot exist."
- "An infinite temporal regress of events is an actual infinite."
- "∴ An infinite temporal regress of events cannot exist."
The second argument, the "argument from the impossibility of completing an actual infinite by successive addition", states:
- "An actual infinite cannot be completed by successive addition."
- "The temporal series of past events has been completed by successive addition."
- "∴ The temporal series of past events cannot be an actual infinite."
Both arguments were adopted by later Christian philosophers and theologians, and the second argument in particular became more famous after it was adopted by Immanuel Kant in his thesis of the first antinomy concerning time.
If we assume that the world has no beginning in time, then up to every given moment an eternity has elapsed, and there has passed away in that world an infinite series of successive states of things. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, and that a beginning of the world is therefore a necessary condition of the world's existence.—Immanuel Kant, First Antinomy, of Space and Time
Modern mathematics generally incorporates infinity, especially after the work of Georg Cantor. Cantor recognized two different kinds of infinity. The first, used in calculus, he called the variable finite, or potential infinite, represented by the sign (known as the lemniscate), and the actual infinite, which Cantor called the "true infinite." His notion of transfinite arithmetic became the standard system for working with infinity within set theory. As David Hilbert said "No one shall drive us from the paradise which Cantor has created for us." However, Hilbert also said that the role of the actual infinite was relegated only to the abstract realm of mathematics. "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought. . .The role that remain for the infinite to play is solely that of an idea." In transfinite arithmetic, inverse operations of subtraction and division with infinite quantities are prohibited because they lead to contradictions. If actual infinites could exist in reality, there would be no way to stop these operations from being performed. Philosopher William Lane Craig argues that if the past were infinitely long, it would entail the existence of actual infinites in reality
Craig and Sinclair also argue that an actual infinite cannot be formed by successive addition. Quite independent of the absurdities arising from an actual infinite number of past events, the formation of an actual infinite has its own problems. For any finite number n, n+1 equals a finite number. An actual infinity has no immediate predecessor.
The Tristram Shandy paradox also illustrates the absurdity of an infinite past. Bertrand Russell asks us to imagine Tristram Shandy, an immortal man who writes his biography so slowly that for every day that he lives, it takes him a year to record that day. Suppose that Shandy had always existed. Since there is a one-to-one correspondence between the number of past days and the number of past years on an infinite past, one could reason that Shandy could write his entire autobiography. From another perspective, Shandy would only get farther and farther behind, and given a past eternity, would be infinitely far behind.
Craig asks us to suppose that we met a man who claims to have been counting down from infinity and is now just finishing. We could ask why he did not finish counting yesterday or the day before, since eternity would have been over by then. In fact for any day in the past, if the man would have finished his countdown by day n, he would have finished his countdown by n-1. It follows that the man could not have finished his countdown at any point in the finite past, since he would have already been done.
The physicist P.C.W. Davies argues that these paradoxes are not mere armchair cosmology. "[T]he universe will eventually die, wallowing, as it were, in its own entropy. This is known among physicists as the 'heat death' of the universe. . .The universe cannot have existed for ever, otherwise it would have reached its equilibrium end state an infinite time ago. Conclusion: the universe did not always exist."
- Seymour Feldman (1967). "Gersonides' Proofs for the Creation of the Universe". Proceedings of the American Academy for Jewish Research (Proceedings of the American Academy for Jewish Research, Vol. 35) 35: 113–137. doi:10.2307/3622478. JSTOR 3622478.
- Craig, William Lane (June 1979). "Whitrow and Popper on the Impossibility of an Infinite Past". The British Journal for the Philosophy of Science 30 (2): 165–170 [165–6]. doi:10.1093/bjps/30.2.165.
- Donald Wayne Viney (1985). "The Cosmological Argument". Charles Hartshorne and the Existence of God. SUNY Press. pp. 65–68. ISBN 0-87395-907-8.
- Herbert A. Davidson (April–June 1969). "John Philoponus as a Source of Medieval Islamic and Jewish Proofs of Creation". Journal of the American Oriental Society (Journal of the American Oriental Society, Vol. 89, No. 2) 89 (2): 357–391. doi:10.2307/596519. JSTOR 596519.
- Mark Daniels (2007). "What's New in Ancient Philosophy". Philosopny Now.[dead link]
- Immanual Kant; Norman Kemp Smith (tr.). "Kant's First Antinomy, of Space and Time". Critique of Pure Reason. pp. A 426–429.
- Philosophy of mathematics : selected readings (Reprint, 2. ed. ed.). Cambridge [u.a.]: Cambridge Univ. Pr. 1991. p. 141. ISBN 978-0521296489.
- Philosophy of mathematics : selected readings (Reprint, 2. ed. ed.). Cambridge [u.a.]: Cambridge Univ. Pr. 1991. p. 151. ISBN 978-0521296489.
- Craig, edited by William Lane; Moreland, J.P. (2011). The Blackwell companion to natural theology ([Pbk. ed.] ed.). Oxford: Wiley-Blackwell. p. 115. ISBN 978-1444350852.
- Craig, edited by William Lane; Moreland, J.P. (2011). The Blackwell companion to natural theology ([Pbk. ed.] ed.). Oxford: Wiley-Blackwell. p. 117. ISBN 978-1444350852.
- Russel, Bertrand (1937). The Principles of Mathematics, 2nd Edition. London: George Allen. p. 358. ISBN 978-0393002492.
- Craig, edited by William Lane; Moreland, J.P. (2011). The Blackwell companion to natural theology ([Pbk. ed.] ed.). Oxford: Wiley-Blackwell. p. 121. ISBN 978-1444350852.
- Craig, edited by William Lane; Moreland, J.P. (2011). The Blackwell companion to natural theology ([Pbk. ed.] ed.). Oxford: Wiley-Blackwell. p. 122. ISBN 978-1444350852.
- Davies, Paul (1984). God and the new physics (1st Touchstone ed. ed.). New York: Simon & Schuster. p. 11. ISBN 978-0671528065.
- Robert Bunn (1988). "Review of Time, Creation, and the Continuum: Theories in Antiquity and the Early Middle Ages by Richard Sorabji". Philosophy of Science 55 (2): 304–306. doi:10.1086/289436.
- Jaakko Hintikka (1966). "Aristotelian Infinity". The Philosophical Review 75 (2): 197–218. doi:10.2307/2183083. JSTOR 2183083.
- Maimonides (1956). The Guide To The Perplexed II. translated by M. Friedlander. London: Dover. pp. 15–16, 25.
- A. W. Moore (2001). "Medieval and Renaissance Thought". The Infinite. Routledge. pp. 46–49. ISBN 0-415-25285-7.
- Richard Sorabji (2005). "Did the Universe have a Beginning?". The Philosophy of the Commentators, 200–600 AD. Cornell University Press. pp. 175–188. ISBN 0-8014-8988-1.
- Ben Waters (2013). "Methuselah’s Diary and the Finitude of the Past". Philosophia Christi 15 (2): 463–469.
- Michael J. White (1992). "Aristotle on Time and Locomotion". The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford University Press. ISBN 0-19-823952-1.