Can the past go back forever?

Has the universe always existed, or did it begin? In The Incoherence of the Philosophers (Tahafut al-Falasifa), al-Ghazali argued that it must have begun, because an actual infinite, a real and completed infinite number of things, cannot exist. If the past had no beginning, an infinite series of events would already have been completed to reach today, which he held to be impossible. For al-Ghazali that beginning pointed to God, who creates by choice; the paper cites the verse “When He decrees a matter, He only says to it, ‘Be,’ and it is” (2:117).

Al-Ghazali illustrated the problem with thought experiments. Imagine two spheres, one turning once a day and one once a year: over an infinite past both would have made infinitely many turns, even though one turns far faster. A chain with no first link could never hang. The paper connects these to Hilbert's Hotel, a modern puzzle in which a hotel with infinitely many rooms, all full, can still take in new guests. Such infinities work in mathematics, the authors note, but lead to absurdities when applied to real events.

The paper weighs three classic objections. Ibn Rushd (Averroes) held that the world could be eternal and still depend on God, as light depends on the sun; the authors reply that this still requires an infinite past to have been crossed, and makes creation automatic rather than willed. David Hume doubted that cause and effect apply to the universe as a whole, and Immanuel Kant held that such reasoning goes beyond any possible experience; the authors answer that modern cosmology does study the universe as a whole. They then turn to science: the expansion of the universe; the Borde–Guth–Vilenkin theorem, which shows that a universe that has on average been expanding cannot be infinitely old; the second law of thermodynamics; and measurements giving the universe an age of about 13.8 billion years.

The authors stop short of calling this a strict proof of God. They conclude that it makes a strong case that the universe is not eternal, and that its beginning points to a necessary cause beyond it.