3.3 Is It Possible to Traverse an Infinite Past?
In the previous chapters, we examined the scientific evidence in favor of a beginning of the universe. Before moving on to the question—if the universe began to exist, what could be the cause of such a beginning?—it is useful to consider another argument in favor of a beginning of time and the universe.
This argument no longer rests on physical data, but on a mathematical and philosophical intuition: the past cannot be infinite, because it would then be impossible to “traverse” that infinity in order to arrive at the present moment.
Potential Infinity vs. Actual Infinity #
It is important to distinguish between two fundamental concepts of infinity:
Potential infinity: a quantity that can grow without limit but remains finite at every moment. For example, counting 1, 2, 3… without ever reaching an “infinite number.” This concept corresponds to the mathematical notion of a limit, where a value approaches infinity without ever actually reaching it.
Actual infinity: an infinite set considered complete and given in its entirety, possessing a proper subset of the same size (such as the natural numbers and the even numbers, which can be placed in one-to-one correspondence). Unlike potential infinity, this kind of infinity is a completed totality, not something built up through finite accumulation.
This distinction is crucial when reflecting on an infinite past: is it a potential infinity (the past stretches back indefinitely without ever being complete), or an actual infinity (the past is a complete and infinite series of events)?
Why an Actual Infinite Past Cannot Be Traversed #
The argument for the impossibility of an infinite past can be stated as follows:
- If time came from an infinite past, then the past would constitute an infinite temporal series of events.
- Such a series of past events would be actually infinite, not merely potentially infinite.
- It is impossible for a series formed by successive addition to be actually infinite.
- But the temporal series of past events has been formed by successive addition (event after event).
- Therefore, time cannot have a beginning infinitely far back in the past.
Conclusions #
If one adopts a dynamic view of time—according to which time really passes—then an infinite past is impossible. Although actual infinity may be conceivable from a static perspective (where all events are given simultaneously, as in a “block universe” view of time), it becomes problematic as soon as one considers the real passage of time.
Even if one grants the existence of actual infinities in reality, the dynamic nature of time prevents an infinite number of events from being actually traversed in order to arrive at the present.
In summary: the real passage of time rules out the possibility of an infinite past, because such an infinity would make the present unreachable. This leads logically to the conclusion that time had an absolute beginning.
Further Reading #
Alan Rhoda, On Traversing an Actually Infinite Past, 2006.
👉 http://alanrhoda.net/blog/2006/05/on-traversing-actually-infinite-past.htmlMorita, Kunihisa. “Time Does Not Pass if Time Began from an Infinite Past”, KRITERION – Journal of Philosophy, vol. 36, no. 3–4, 2022, pp. 291–302.
👉 https://doi.org/10.1515/krt-2022-0001Aristotle, Physics, Book III: distinction between actual infinity and potential infinity.