Here are two arguments that an infinite past is logically impossible, and why they’re wrong.
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
Note, 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. 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:
- if the universe was temporally infinite, then there was no 1st moment
- if there was no 1st moment (but just some moment), then there was no 2nd moment
- if there was no 2nd moment (but just some other moment), then there was no 3rd moment
- ... and so on and so forth ...
- if there was no 2nd last moment, then there would be no now
- since now exists, we started out wrong, i.e. the universe is not temporally infinite
Seems convincing at a glance?
In short, the argument (merely) shows that, on an infinite temporal past, the now cannot 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. That is, the latter (conclusion) is a
non sequitur, and the latter two could be expressed more accurately as:
- if there was no 2nd last moment with an absolute number, then there would be no now with an absolute number
- since now exists, we started out wrong, i.e. any now does not have an absolute number
Additionally, note that 1,2,3 refer to non-indexical “absolute” moments (1
st, 2
nd, 3
rd), but the following steps are indexical and contextual (2
nd last, now), which is masked by “... and so on and so forth ...”. We already know from elsewhere (originating in linguistics) that such reasoning is problematic.
Still no proof, as some of the religious apologists propose.
:-|