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In my previous post, I critically examined an article by Ben Waters, in which he used a story of Methuselah and his diary in order to argue for the finitude of the past.  In particular, I showed that the argument was inherently circular.  For more details, including the full statement of Waters’ argument, see here:

The purpose of the current post is to critique Waters’ argument in a different way.  Rather than show precisely what is wrong with the argument (which I did in the previous post), I’ll show that the argument may be modified in order to prove something else, something that Waters presumably disagrees with – namely, that the future must be finite.

This style of argumentation (applying the logic of an argument to prove something else) should be familiar to anyone who has read discussions of ontological arguments; see for example my post Resurrecting Victor Stenger’s Ontological Pizza.  See also Wes Morriston’s excellent (and freely available) article Beginningless past, endless future, and the actual infinite, where he applies this idea to several other arguments for a finite past.

This type of criticism is sometimes thought of as being a bit weak, as all it shows is that if one accepts Argument A, then one ought also to accept Argument B.  This does not automatically tell us that Argument A is flawed (perhaps we should accept the conclusions of both arguments), but it does alert someone who rejects the conclusion of Argument B to be wary about Argument A.  In any case, I think it is entirely appropriate in this case:  not only is it a bit of fun, but the Tristram Shandy paradox (on which the Methuselah paradox is based) came about when William Lane Craig turned a famous story of Tristram Shandy on its head – so why not return the favour?

I quoted Waters’ argument in full in my previous post (linked above), so I will not do so again.  Instead, I’ll just present my own adaptation of the argument.  So, without further ado, I give you…

God’s Diary – an argument for the finitude of the future

I first introduce some notation.  Let t represent today, D the collection of all days that will occur after t, ≤ the total relation on D where d1 ≤ d2 if and only if d1 is earlier than or the same as d2, and = the relation on D where d1 = d2 if and only if d1 is the same as d2.  Furthermore, for any d in D and any positive integer n such that d follows t by at least (n+1) days let (d-n) represent the unique nth day preceding d in D; similarly, if d does not precede a possible final day of D by less than n days let (d+n) represent the unique nth day following d in D.  Note that if d is not a final day of D then it has a following day (d+1) in D.  Finally, let DF represent the sub-collection of all days in D that are finitely distant in the future so that DF only contains those days in D that are of the form (t+m) for some positive integer m and not any possible days in D that are infinitely distant in the future.

With these conventions in mind, the argument is given as follows:

(1)  If there exists a function f from DF to DF such that d ≤ f(d) for any d in DF and f(d-2) = (f(d)-1) for any pair d, (d-2) in DF then D is finite.

(2)  There exists a function f from DF to DF such that d ≤ f(d) for any d in DF and f(d-2) = (f(d)-1) for any pair d, (d-2) in DF.

(3)  Therefore, D is finite by (1) and (2).

I establish (1) by first noticing that DF is either a finite collection with a final day or the infinite collection { (t+n) | n is any positive integer } and assume that there exists a function f from DF to DF such that d ≤ f(d) for any d in DF and f(d-2) = (f(d)-1) for any pair d, (d-2) in DF.  Suppose DF can be identified with the aforementioned infinite collection and observe that f(t+1) = (t+m) for some positive integer m.  It follows that (t+m) = f(t+1) = f((t+(1+2m))-2m) = (f(t+(1+2m))-m) so that f(t+(1+2m)) = (t+2m), which contradicts the fact that (t+(1+2m)) ≤ f(t+(1+2m)).  Hence, DF cannot be the aforementioned infinite collection so that it is a finite collection with a final day e.  Now, if e is not also a final day of D then it has a following day (e+1) in DF, which contradicts the fact that e is the final day of DF.  Hence, e is not only the final day of DF but also the final day of D.  However, if D has a final day that is finitely distant in the future then it follows that D is finite so that (1) is established.

I now establish (2) by considering the case of God, who will be alive for every d in DF. More pertinently, God plans to maintain a diary of his life.  However, recording the past is no fun to a maximally great being, so God plans to write about days that have not yet come to pass.  God only works on entries for his diary in the mornings and never on more than one entry per day, with each entry summarizing his future activities for some d in DF. Additionally, for any pair d, (d+1) in DF God has a perfect knowledge on d of everything he will do on (d+1) and whenever he works on an entry for his diary it is always at a rate of half an entry per day. With these stipulations in mind, God works on entries for his diary in the following manner:

For any d in DF, if God foresees that he will be working on an entry for (d+m) on the following day (d+1) for some positive integer m then in the morning of day d he will work on his diary in such a way that the entry he will write on day (d+1) will continue on from the entry he writes on day d.  In particular, if God foresees that he will write about the evening of day (d+m) on day (d+1) then he will write about the morning of day (d+m) on day d, otherwise he will write about the evening of day (d+m-1) on day d.  On the other hand, if God foresees that he will not work on such an entry on a following day (d+1) then in the morning of day d he will write about the evening of day d.  Note that this would include the case where d does not have a following day (d+1) in D.

Finally, I remark that God is thought of as belonging to a logically possible world that shares all the relevant temporal facts with the actual world.

It follows that for any d in DF God will work on an entry for either d or one of (d+m), (d+m-1) for some positive integer m, but in any case he will work on an entry for a day in DF that is later than or the same as d.  Hence, there exists a function f from DF to DF such that f(d1) = d2 if and only if God works on an entry for d2 on d1 for any d1, d2 in DF with d ≤ f(d) for any d in DF.  Finally, I observe that f(d-2) = (f(d)-1) for any pair d, (d-2) in DF since God always takes two consecutive days to finish writing an entry for his diary and he always works on subsequent days finishing where he will begin at the beginning of following days given his foreknowledge and the manner in which he works on entries for his diary in the morning of every d in DF.  Hence, f satisfies all the conditions needed to establish (2).

Conclusion

Premises (1) and (2) have been established, and the conclusion (3) follows by a simple application of the modus ponens law.  So the above argument seems to show that the future is necessarily finite.  In other words, the very idea of the world (or even God) lasting forever is logically incoherent!

Certainly anyone who believes otherwise will suspect there must be a problem with the logic of the above argument.  And I agree with them!  I know exactly what the problem is, and it is precisely the same problem as Waters’ original argument for the finitude of the past.  If you can refute my current argument, you can refute Waters’ argument (and vice versa).  So, instead of spoiling the fun, I’ll leave the refutation of my argument as an exercise for the reader… Category: AtheismBen WatersCosmological argumentInfinityTheismTime ### Article by: Reasonably Faithless

Mathematician and former Christian