Here are two of the typical arguments against an infinite past, and why they don't hold up. (Some basic mathematics required.)
Last Thursdayism
- Assumption (towards reductio ad absurdum): infinite temporal past
- Let's enumerate past days up to and including last Wednesday as: {..., t, ..., -1, 0}
That is, there exists a bijection among those past days (including Wednesday) and the non-positive integers
- Now come Thursday
- Observation: {..., t, ..., -1, 0} cannot accommodate Thursday
- Let's re-enumerate the same past days but including Thursday as: {..., t, ..., -1, 0}
That is, there exists a bijection among those past days (including Thursday) and the non-positive integers
- Observation: {..., t, ..., -1, 0} can accommodate Thursday
- The two observations are contradictory
- {..., t, ..., -1, 0} both cannot and can accommodate Thursday
- Conclusion: the assumption is wrong, an infinite past is impossible
This argument could equally be applied to infinite causal chains, and nicely lends support to the
Omphalos hypothesis (hence why I named it
Last Thursdayism). Another thing to notice about the infinite set of integers: any two numbers are separated by a number. And this number is also a member of the integers. That is,
the integers are closed under subtraction and addition. For the analogy with enumerating past days, this means any two events are separated by a number of days. Not infinite, but a particular number of (possibly fractional) days. That's any two events. To some folk this is counter-intuitive, but, anyway, there you have it.
The first observation is incorrect. Whether or not the set can accommodate Thursday (one more day), is not dependent on one specific
bijection (the first selected), rather it is dependent on the existence of some (any such)
bijection. A
bijection also exists among {..., t, ..., -1, 0} and {..., t, ..., -1, 0, 1}, and the integers, for that matter.
Therefore, the argument is not valid.
The unnumbered now
1. if the universe was temporally infinite, then there was no 1
st moment
2. if there was no 1st moment (but just some moment), then there was no 2
nd moment
3. if there was no 2nd moment (but just some other moment), then there was no 3
rd moment
4. ... and so on and so forth ...
5. if there was no 2
nd last moment, then there would be no now
6. since now exists, we started out wrong, i.e. the universe is not temporally infinite
The argument shows that, on an infinite temporal past, the now doesn't have a definite, specific number, as per 1
st, 2
nd, 3
rd, ..., now. Yet, we already knew this in case of an infinite temporal past, so, by implicitly assuming otherwise, the argument can be charged with
petitio principii.
Additionally, note that 1,2,3 refer to non-indexical "absolute" moments (1
st, 2
nd, 3
rd), but
5 is indexical and contextual (2
nd last, now), which is masked by
4. We already know from elsewhere (originating in linguistics) that such reasoning is problematic.
That is,
6 is a
non sequitur, and could be expressed more accurately as:
5. if there was no 2
nd last moment with an absolute number, then there would be no now with an absolute number
6. since now exists, we started out wrong, i.e. any now does not have an absolute number
Hilbert's Hotel and
Shandy's Diary, for example, are peripherally related, known veridical paradoxes, and do not imply a contradiction, but they do show some counter-intuitive implications of infinites.
However, completing an infinite process is not a matter of starting at a particular time that just happens to be infinitely far to the past and then stopping in the present. It’s to have always been doing something and then stopping. This point is illustrated by a possibly apocryphal story attributed to the philosopher Ludwig Wittgenstein. Imagine meeting a woman in the street who says, “Five, one, four, one, dot, three! Finally finished!” When we ask what is finished, she tells us that she just finished counting down the infinite digits of pi backward. When we ask when she started, she tells us that she never started, she has always been doing it. The point of the story seems to be that impossibility of completing such an infinite process is an illusion created by our insistence that every process has a beginning. — James Harrington
There is no logical or conceptual barrier to the notion of infinite past time.
In a lecture Wittgenstein told how he overheard a man saying '...5, 1, 4, 1, 3, finished'. He asked what the man had been doing.
'Reciting the digits of Pi backward' was the reply. 'When did you start?' Puzzled look. 'How could I start. That would mean beginning with the last digit, and there is no such digit. I never started. I've been counting down from all eternity'.
Strange, but not logically impossible. — Craig Skinner
- Pathways to Philosophy - Ask a Philosopher: Questions and Answers 47 (2nd series), question 94
∞ does not derive a contradiction, rather, to learn more about our world, we'll have to go by evidence and try to piece things together.
Whitrow and Popper on the impossibility of an infinite past by William Lane Craig
Georg Cantor (1845-1918): The man who tamed infinity by Eric Schechter