• Methuselah’s Diary – a response to Ben Waters on the finitude of the past

     

    1.  Summary

    This post examines an article by Ben Waters, entitled Methuselah’s Diary and the Finitude of the Past, that has recently appeared in the journal Philosophia Christi.  Waters’ article concerns a certain type of argument against an infinite past that I’ll call a “Shandy-style argument” (see below for more details).  In his article, Waters surveys various Shandy-style arguments given in the past by other philosophers, explains what he sees as their flaws, and then provides his own new Shandy-style argument, which he believes fixes the problems faced by the old ones, and thereby successfully demonstrates the impossibility of an infinite past.

    The main goal of this post is to explain exactly what I think is wrong with Waters’ argument.  In particular, I’ll show that his argument begs the question (that is, it assumes the conclusion right from the beginning).  Moreover, I’ll show that the reasoning he gives in support of one of his premises may be adapted to prove that the sub-argument he gives to support his other premise is faulty (unless one already assumes the conclusion).

    UPDATE:  There is now a second part to the critique of Waters’ argument:

    I’ve written about other arguments regarding the finitude/infinitude of the past in the following posts (and several others):

    The current post should be considered as the next instalment of a probably very long series.  When I have some more time, I hope to rewrite parts of this post into a journal article – I’ll link to that when it happens.  I should also say at the outset that some parts of this post are a little heavy on logic and mathematics – most of my posts are not like this, but a proper treatment of Waters’ argument necessitates this kind of approach.

    2.  Introduction and background

    The Life and Opinions of Tristram Shandy, Gentleman is an 18th Century classic book by Laurence Sterne.  The Wikipedia page on Tristram Shandy contains some background information about the book but, for the present discussion, the main item of importance is this:

    Shandy is writing his own autobiography, but he is so meticulous that it takes an entire year for him to record a single day.

    So as Shandy works through his autobiography, he gets further and further behind.  Indeed, Shandy would be a whopping 365 years old when he finishes writing about just the first year of his life – and that’s assuming he is born with the ability to write.  Obviously, whenever Shandy dies, he will not have completed his autobiography – not by a long shot.

    However, as Bertrand Russell noted in his famous book, Principles of Mathematics, if Shandy lived forever, every day of his life would eventually be recorded.  Indeed, the entry about his 100th birthday would be available for reading at approximately the end of his 36,500th year of life – Shandy must have a pretty good memory!  Even though Shandy will never actually finish writing his autobiography, and even though he gets further and further behind in his task, no day would go unrecorded.

    William Lane Craig has turned the situation around in an effort to argue against the possibility of an infinite past; for example, see his book The Kalam Cosmological Argument.  The idea is that if Shandy had been alive from eternity past, and had been writing his autobiography all the time at a rate of one day recorded for every year of his life, then certain difficulties would allegedly arise that (according to Craig) demonstrate the absurdity of an infinite past.  For the sake of completeness, I’ll include Craig’s argument in the next section.

    3.  Craig’s argument

    What follows is William Lane Craig’s original version of the Tristram Shandy paradox, and is taken from his above-mentioned book, as quoted in Waters’ article:

    [S]uppose Tristram Shandy has been writing from eternity past at the rate of one day per year. Would he now be penning his final page? Here we discern the bankruptcy of the principle of correspondence in the world of the real. For according to that principle, Russell’s conclusion would be correct: a one-to-one correspondence between days and years could be established so that given an actual infinite number of years, the book will be completed. But such a conclusion is clearly ridiculous, for Tristram Shandy could not yet have written today’s events down. In reality he could never finish, for every day of writing generates another year of work. But if the principle of correspondence were descriptive of the real world, he should have finished — which is impossible.

    …But now a deeper absurdity bursts into view. For if the series of past events is an actual infinite, then we may ask, why did Tristram Shandy not finish his autobiography yesterday or the day before, since by then an infinite series of events had already elapsed? No matter how far along the series of past events one regresses, Tristram Shandy would have already completed his autobiography. Therefore, at no point in the infinite series of past events could he be finishing the book. We could never look over Tristram Shandy’s shoulder to see if he were now writing the last page. For at any point an actual infinite sequence of events would have transpired and the book would have already been completed. Thus, at no time in eternity will we find Tristram Shandy writing, which is absurd, since we supposed him to be writing from eternity. And at no point will he finish the book, which is equally absurd, because for the book to be completed he must at some point have finished. What the Tristram Shandy story really tells us is that an actually infinite temporal regress is absurd.

    I don’t really wish to comment at length on this, as I think that (with all due respect) almost all of what Craig says there is complete nonsense – Waters rightly observes that it is difficult to ascertain exactly what Craig is even trying to say.  In any case, Craig’s argument has been well critiqued by several authors, including Quentin Smith (see Part IV of his 1987 article Infinity and the Past in the journal Philosophy of Science), Ellery Eells (see his 1988 article Quentin Smith on Infinity and the Past in the same journal), and Wes Morriston (in his outstanding 1999 Philo article, Must the Past Have a Beginning? – this article is freely available at the link just provided and is highly recommended reading on this topic).  Waters gives the details of several other critiques of Craig’s argument in his article.  But, most importantly, Waters echoes the thoughts of Morriston (from his above-mentioned article) that to use the Tristram Shandy story to demonstrate that the past is necessarily finite, one would have to argue along the following lines:

    (1)  If the past were infinite then it is possible that Tristram Shandy could have been recording consecutive past days at a rate of one day per year from eternity past.

    (2)  It is not possible that Tristram Shandy could have been recording consecutive past days at a rate of one day per year from eternity past.

    (3)  Therefore, the past is not infinite by (1) and (2).

    And, as Waters notes:

    the more significant difficulty with Craig’s argument, or a similar argument that proceeded along the lines of Small’s analysis [RF note: see Waters’ article for details on Small’s analysis], is that even if it succeeds in demonstrating that Tristram Shandy could not possibly have been recording consecutive past days at a rate of one day per year from eternity past it still remains to show (1), which is to say that Tristram Shandy should have been able to have done so given an infinite past.

    As it turns out, Waters’ new argument (see the next section) suffers from precisely the same criticism, though we have to dig a little deeper to realise this.  The benefit of Waters’ treatment, however, is that he sets up the argument in a far more formal manner, and this makes it much easier to explain precisely what is wrong with it.  I believe that the critiques I’ll level against Waters’ argument are likely to work against any kind of Shandy-style argument, but this belief could only be tested on a case-by-case basis as future arguments are made.  So, without further ado, let’s turn to Waters’ argument.

    4.  Waters’ argument

    Here I’ll quote the important section of Waters’ paper (Section II, entitled Methuselah’s Diary), in which he articulates his argument and then gives his support to its premises.  For simplicity, I’ve written subscripts inline so, for example, d1 will be rendered as d1, and so on.  Please read Waters’ paper for more details – the version I’m quoting from is the freely available one I’ve just linked to.

    I first introduce some notation. Let t represent today, D the collection of all days prior to 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 ≤ d2 and d2 ≤ d1 (i.e., d1 is the same as d2). Furthermore, for any d in D and any positive integer n such that d precedes t by at least (n+1) days let (d+n) represent the unique nth day following d in D; similarly, if d does not follow a possible earliest day of D by less than n days let (d-n) represent the unique nth day preceding d in D. [Footnote: Note that if d is not an earliest day of D then it has a previous day (d-1) in D.] Finally, let DF represent the sub-collection of all days in D that are finitely distant in the past 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 past.

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

    (1)  If there exists a function f from DF to DF such that f(d) ≤ 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 f(d) ≤ 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 an earliest 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 f(d) ≤ 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 f(t-(1+2m)) ≤ (t-(1+2m)). Hence, DF cannot be the aforementioned infinite collection so that it is a finite collection with an earliest day e. Now, if e is not also an earliest day of D then it has a previous day (e-1) in DF, which contradicts the fact that e is the earliest day of DF. Hence, e is not only the earliest day of DF but also the earliest day of D. However, if D has an earliest day that is finitely distant in the past then it follows that D is finite so that (1) is established.

    I now establish (2) by considering the case of Methuselah, who has been alive for every d in DF and is the oldest living individual. More pertinently, Methuselah has maintained a diary of his previous activities throughout his long life; however, he only works on entries for his diary in the evenings and never on more than one entry per day, with each entry summarizing his past activities for some d in DF. Additionally, for any pair (d-1), d in DF Methuselah has a perfect memory on d of everything he did 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, Methuselah works on entries for his diary in the following manner [RF note: emphasis has been added by me]:

    For any d in DF, if Methuselah remembers working on an entry for (d-m) on a previous day (d-1) for some positive integer m then in the evening he will continue working on his diary where he had left off at the end of (d-1). In particular, if Methuselah had not finished working on an entry for (d-m) then he will continue working on that entry, otherwise he will start working on an entry for ((d-m)+1). On the other hand, if Methuselah does not remember working on such an entry on a previous day (d-1) then in the evening he will start working on an entry for d. [Footnote: This would include the case where d does not have a previous day (d-1) in D.]

    Finally, I remark that Methuselah 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 Methuselah 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 earlier 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 Methuselah works on an entry for d2 on d1 for any d1, d2 in DF with f(d) ≤ 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 Methuselah always takes two consecutive days to finish writing an entry for his diary and he always works on subsequent days where he had left off at the end of previous days given his memory and the manner in which he works on entries for his diary in the evening of every d in DF. Hence, f satisfies all the conditions needed to establish (2).

    (For those who don’t know, Methuselah is a character from the Old Testament who is said to have lived for 969 years, longer than anyone else in the Bible; see Genesis 5:21-27.)

    5.  My critique of Waters’ argument

    In this section, I’ll explain the flaw I see in Waters’ argument.  In what follows, I will use the same symbols to refer to days and the ≤ ordering on them, and I will use the notation D and DF as Waters used them.  To simplify matters I will say that a function f from DF to DF has:

    Property 1  if f(d) ≤ d for any d in DF, and

    Property 2  if f(d+2) = (f(d)+1) for any pair d, (d+2) in DF.

    With these conventions, Waters’ argument may be restated more simply as:

    (1)  If there exists a function f from DF to DF with Properties 1 and 2, then D is finite.

    (2)  There exists a function f from DF to DF with Properties 1 and 2.

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

    Before explaining what I think is wrong with the argument, let me state my points of agreement with Waters.  First, note that the two premises (1) and (2) logically entail the conclusion (3); we have a straightforward application of the modus ponens rule.  So the value of the argument rests entirely on whether the premises (1) and (2) are true.  I accept (1) as true – note that this premise is essentially a mathematical proposition, and Waters gave a perfectly good mathematical proof of it (see above).  The real problem I have with the argument is with premise (2) or, more specifically, with the argument given in its support, and the rest of this section is devoted to explaining the problem.

    In a nutshell, my problem with Waters’ support for premise (2) is as follows.  In order to support (2), Waters makes use of a story about Methuselah; this story is in bold in the previous section, and I’ll refer to it as the Methuselah Story.  In particular, Waters imagines Methuselah writing in his diary in a certain way, and then uses the details of the story to construct a function with Properties 1 and 2.  Since (2) is a premise in an argument whose conclusion (3) is that D (the collection of all past days) is finite, any support of (2) should not depend on the assumption that D is finite; otherwise, the argument would be circular, and entirely worthless.  In other words, Waters really needs to establish that, regardless of whether D is finite or infinite, (2) ought to be true.  (Compare with the last quote from Waters in Section 3.)  But the problem with the Methuselah Story is precisely that it is logically incoherent if D is infinite, as I’ll now explain.  It will be convenient to break the argument that follows into three separate propositions and their corresponding proofs.

    Proposition 1.  If D is infinite, then DF is infinite also.

    Proof.  Waters showed above that the statement “DF is finite” implies the statement “D is finite”.  Proposition 1 is therefore true, as it is simply the contrapositive of Waters’ implication.  ☐

    Proposition 2.  If the Methuselah Story is logically coherent then, regardless of whether D is finite or infinite, on any day in DF, Methuselah wrote in his diary about some day from DF.

    Proof.  Consider an arbitrary day d from DF.  According to the story, if on day d Methuselah remembers writing in his diary on day (d-1), say about day (d-m) from DF, then Methuselah will write about either day (d-m) or day (d-m+1) on day d.  On the other hand, if Methuselah does not remember writing about any day on day (d-1), then he will write about day d on day d.  In any case, on day d, Methuselah writes about some day from DF.  Since d was an arbitrary day from DF, the proposition has been proven.  ☐

    In order to proceed, we must state a crucial assumption:

    Assumption A.  Methuselah cannot write about a day that has not yet occurred (at the time of writing).

    Note.  In the next section, I’ll discuss Assumption A in more detail.  For now, let’s just say that it seems pretty reasonable to suppose Methuselah cannot predict the future.  As we’ll see in the next section, however, denying Assumption A leads to a different kind of problem with Waters’ argument.

    Proposition 3.  If D is infinite, the Methuselah Story is logically incoherent.

    Proof.  In order to obtain a contradiction, suppose D is infinite and that the Methuselah Story is logically coherent.  Since D is assumed to be infinite, it follows by Proposition 1 that DF is infinite also.  Thus, DF = { (t-n) | n is any positive integer }.  Now consider Methuselah’s activities on day (t-1), ie the day before today.  By Proposition 2, on day (t-1), he wrote about day (t-m) for some positive integer m.  It follows that two days before (t-1), ie on day (t-3), Methuselah wrote about day (t-m-1).  Two days before that, ie on day (t-5), he wrote about day (t-m-2).  Two days before that, ie on day (t-7), he wrote about day (t-m-3).  Continuing in this fashion, we see that for any positive integer k, on day (t-2k-1), Methuselah wrote about day (t-m-k).  In particular, when k = m, it follows that on day (t-2m-1), Methuselah wrote about day (t-m-m) = (t-2m).  But day (t-2m) is the day after day (t-2m-1).  In other words, on day (t-2m-1), Methuselah wrote about a day that had not yet occurred.  But this contradicts Assumption A.  Proposition 3 has therefore been established.  ☐

    To summarise where we have arrived at, it has been shown that the Methuselah Story is only logically coherent if it is also assumed that D is finite.  In other words, to support one of the premises of his argument, Waters makes use of a sub-argument that relies crucially on the conclusion of his argument.  That is, the argument is circular.

    6.  Remarks about my critique

    First, if anyone felt some deja vu while reading my proof of Proposition 3, this is because it is based very closely on the argument Waters gave in support for his premise (1).

    Second, someone might be uneasy about my use of the phrase “Continuing in this fashion” in the proof of Proposition 3, since it implicitly assumes the Axiom of Induction.  To this, I simply remark that Waters also (implicitly) made use of the same axiom in the corresponding part of his defense of premise (1).  Namely, he needs the axiom to deduce that f((t-(1+2m)) +2m) = (f(t-(1+2m))+m) for any function f satisfying Property 2.

    But, of course, the most interesting consideration is the role of Assumption A in the proof of Proposition 3, as it was this assumption that gave us the crucial contradiction.  So it must be asked if there are good reasons to accept Assumption A.  Well, it seems perfectly acceptable to me (not that that should be considered a satisfactory reason).  There also seem to be physical restrictions on our universe that prevent us discovering information about the future.  But let’s leave such issues aside, since perhaps we could assume Methuselah has the power to foresee the future, or that the story could be re-told with the role of Methuselah played by a supernatural being (such as God – leaving aside the issue that some theologians deny that God has perfect knowledge of the future).  The biggest problem for anyone who might want to deny Assumption A in order to invalidate my proof of Proposition 3 is that Waters uses Assumption A (albeit implicitly) in his telling of the Methuselah Story.  In particular, Waters stipulates that if, on day d, Methuselah recalls writing in his diary on day (d-1) about some day d’, then he will write about either day d’ or day (d’+1) on day d.  However, Waters assumed that day d’ must have been day (d-m) for some positive integer m.  In other words, Waters assumes that if Methuselah wrote in his diary on day (d-1), then he did not write about a day in the future (from the perspective of day (d-1)).  If we deny Assumption A (or just don’t assume it), then we remove all warrant to assume that d’ = (d-m) for some positive integer m.  The effect of all this is that (regardless of whether D is finite or infinite), we cannot deduce that the function f obtained from the Methuselah Story in the way Waters describes satisfies Property 1, ie that f(d) ≤ d for any d in DF.

    In other words, the denial of Assumption A leaves us with a logically coherent version of the Methuselah Story (or at least it removes the force of the objection I have raised – there may be other reasons to think the story is incoherent), but this version does not establish premise (2).  Therefore, the denial of Assumption A only serves to replace one fatal flaw with Waters’ argument with another.

    7.  Could a Shandy-style argument ever work?

    I’ve already said that I accept Waters’ first premise:

    (1)  If there exists a function f from DF to DF with Properties 1 and 2, then D is finite.

    But now consider the following statement, which is the converse of (1):

    (1)’  If D is finite, then there exists a function f from DF to DF with Properties 1 and 2.

    This statement is also true.  Indeed, suppose D is finite.  Then D = DF, and there must have been a first day, say day e.  Suppose that yesterday, ie day (t-1), is the nth day after e, ie (t-1) = (e+n).  It follows that D = DF = {e, (e+1), (e+2), … , (e+n) }.  If we allow ourselves to write e = (e+0), then we see that every day from D is of the form (e+k) where k is some integer from {0, 1, 2, … , n}.  We may then define a function f from DF to DF by f(e+k) = (e+⌊k/2⌋) for each k in {1, 2, … , n}.  Here, ⌊k/2⌋ denotes the largest integer not greater than k/2 (this is just the standard floor function, even though the symbols look a little strange in this font).  For example, if k = 7, then ⌊k/2⌋ = ⌊7/2⌋ = ⌊3.5⌋ = 3, while if k = 8, then ⌊k/2⌋ = ⌊8/2⌋ = ⌊4⌋ = 4.

    Note that if k is any non-negative integer, then ⌊k/2⌋ ≤ k/2 ≤ k.  It follows that if d is a day from DF, then d = e+k for some k in {0, 1, 2, … , n}, and so f(d) = f(e+k) = (e+⌊k/2⌋) ≤ (e+k) = d.  That is, the function f satisfies Property 1.

    Also, if d, (d+2) is any pair of days from DF, then d = (e+k) and (d+2) = (e+k+2) for some k in {0, 1, 2, … , n-2}, and so f(d+2) = f(e+k+2) = (e+⌊(k+2)/2⌋)= (e+⌊k/2 + 1⌋) = (e+⌊k/2⌋+1) = (f(d)+1).  That is, the function f satisfies Property 2.

    So statement (1)’ has been verified.  (I remark that the function f, so defined, is actually the function obtained from the Methuselah Story, as per Waters’ recipe, in the case of a finite past.  In fact, the function f, so defined, is one of only two functions from DF to DF with Properties 1 and 2; the other is defined by f(e+k) = (e+⌊(k+1)/2⌋) for each k in {1, 2, … , n}.)

    Statements (1) and (1)’ may be put together to yield the rather interesting (and true) statement:

    (1)”  D is finite if and only if there exists a function f from DF to DF with Properties 1 and 2.

    I find it fascinating that the statement “the past is finite” is logically equivalent to a statement concerning the existence of a certain kind of function.  However, I suppose it is not exactly a surprise, given that any number of similar statements could be given.  For example, the following statement is also true (the proof is left as an exercise for the reader – let me know if you figure it out):

    (1)”’  D is infinite if and only if there exists a function f from DF to DF such that f(d) < d for all d in DF.

    But the important lesson to take from statement (1)” is that premise (2) from Waters’ argument is precisely as strong a statement as the conclusion (3), which states that the past is finite.  So if premise (2) is to be used in an argument for the finitude of the past, then it will need to be supported with an argument that is already strong enough to support the conclusion.  For this reason, I suspect that arguments along the lines of Waters’ one are of no use in establishing the finitude of the past.  I suspect the same can be said for any Shandy-style argument, since they all seem to work in the same kind of way.  Of course, I will be happy to be corrected on this if someone discovers a reason to think otherwise.

    8.  Conclusion

    I have not proven (nor have I attempted to prove) that the past is infinite; rather, I have proven that a certain argument offered by Waters does not succeed in proving that the past is finite.

    Even in doing this, I have not proven (nor have I attempted to prove) that a premise in Waters’ argument is false; rather, I have proven that Waters has not adequately supported premise (2) of his argument.  It might be possible for Waters, or someone else, to provide a different argument (one that does not assume the past is finite) to show that there should be a function satisfying Properties 1 and 2.  I suspect this is an impossible task, but I will be happy to evaluate any attempt to undertake it.

    So, as I have said before, I do not take any of the above discussion to support a claim to the effect of the past being infinite.  But I do take it to show that Waters has not given us any reason to suppose the past is finite.  It seems to me that it is appropriate to remain agnostic about the finitude or infinitude of the past, and that is where I shall stay for the time being.

    methuselah

     

     

     

    Category: Ben WatersCosmological argumentInfinityTimeWilliam Lane Craig

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    Article by: Reasonably Faithless

    Mathematician and former Christian
    • I agree with everything you said here. In fact, I was already agreeing by the end of section 4. And while you don’t offer any negative judgment on Waters, I offer my opinion that his argument is sloppy, and the hidden assumption should have been obvious to him.

      I am far more pessimistic that any Tristram Shandy argument could work, or indeed any a priori argument against an infinite past. I think if such an argument did exist, it would amount to proving the inconsistency of arithmetic (and therefore also of ZFC set theory). I mean, we can’t prove arithmetic is consistent, but attempting to prove its inconsistency is certainly quixotic.

      • I agree very much with your last paragraph – I’ve said in a few places that I doubt an argument against an infinite past could work. I’m prepared to be corrected, of course, since I don’t claim to have a proof that no such argument could be successful. But I’m quite pessimistic (that’s a good way to put it), since all the arguments so far seem so weak. Most of them appear to just point out differences between finite and infinite arithmetic – Waters had a great quote in his article from David Conway: “[other] arguments for the crucial premise that an infinite regress is impossible do little more than point out that infinite series have odd properties”.

        Speaking of Waters, I think he’s a good guy. I’ve only communicated via email (he said he liked my previous paper, and thought it did a good job of correcting Craig’s argument based on inverse operations with infinite arithmetic). I think the argument Waters presented has its problems, but it is far less sloppy than many other Shandy-style arguments I’ve seen. It raised the bar, even though it can still be refuted. It’s hard to know whether he suspected there was a problem with it or not, and I try not to go into that meta-argument. I’m hoping he’ll share his own thoughts on my critique, and I’m expecting we’ll have a good discussion.

        By the way, are you a knot theorist? I’ve done a bit of work on braid groups and monoids – it’s not a million miles away…

        • No, I’m an experimental physicist. I just like math.

          I agree with your point about raising the bar. I definitely appreciated Waters’ proof of the first premise.

    • josh

      These Shandy-type arguments seem really pointless and needlessly complicated in Waters attempted proof above. There are infinite series with a ‘beginning’ (e.g. the positive integers) and those without (the integers). If you make up a story about someone who is in the middle of a series with a beginning, you find that that it has a first element. This cannot possibly prove that time is such a series, since there is nothing logically problematic about an open ended infinite series. As you say, Shandy/Methuselah stories assume a description that cannot apply to an infinite past, which tells us nothing about what the actual past is like.

      I marvel at people like Waters and Craig who clearly lack mathematical training but feel confident to advance ‘proofs’ like this. For instance: “Finally,
      let DF represent the sub-collection of all days in D that are finitely
      distant in the past 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 past.”

      All days in D are finitely distant in the past and can be notated as t-m. There is no such thing as a day which is infinitely distant, just as there is no number which is infinity, the collection of individual days is infinite.

      • Based on our intuition about how days work, the sets D and DF are equivalent. But mathematically speaking, it’s possible that D and DF are distinct. For example, consider D = {t-m | m is a positive integer} U {e+n | n is a non-negative integer}, where all days of the form e+n are defined to be earlier than all days of the form t-m. This technically obeys all our assumptions about D, although as a real timeline we would find it quite strange.

        In a sense, you’re correct that distinguishing D and DF is needlessly complicating, because they could have simply taken an additional assumption (ie D = DF). But they didn’t need that assumption, so they didn’t assume it.

        • josh

          I didn’t say one has to define D and DF as equivalent. I’m saying that as defined in the article they are. If D is a finite set then clearly all its members are in DF. But if D is infinite it is still true that all specific days are a finite distance from t. There is no such thing as a day or days that are infinitely in the past.

          I take it in your example that ‘e’ is supposed to represent negative infinity or some such, but e+n does not provide a unique number to every member of the set. It is improperly defined. For any given day, it could be written as t-m so it isn’t possible to define a set of days which are all earlier than the set which can be written as t-m. All negative integers are of the form t-m.

          • Thanks for your comments and thoughts, Josh.

            Regarding the distinction between D and DF, Waters was actually being very careful. If you read some of the Shandy-style arguments (or just papers about philosophy of time in general), you’ll find several people talking about infinitely distant days. Waters was just being careful to note that, whether such days do or don’t exist (under an assumption of an infinite past), his argument is not going to be defeated by someone raising them as an objection. I think he handled the situation expertly!

            trivialknot explained the situation quite well. Another way to put it is that just assuming that the total *number* of past days is (countably) infinite does not give us any warrant to assume that they have a certain *order*. For example, we can order the set of positive integers as 1,2,3,4,5,6,… or 1,3,5,…,2,4,6,… – note that in the second ordering, there are infinitely many numbers in the sequence *before* 2 – and yet there is still a “first element” in the list. We could also order the integers as 1,3,5,…,…,6,4,2. Here again, there are infinitely many numbers in the list, but there is a first and last element. It is conceivable that an infinite past could have some strange order properties, though it’s hard to wrap my head around it. The same kind of thing happens when you consider super-tasks.

            • josh

              Thanks for your reply. What you have done in the evens and odds example is break an infinite set into two infinite sets (which of course you can do in this example). However, ‘ordering’ has now taken on a different meaning. You say ‘I have one ordered set, then another’ but or course it is trivial to reverse the order in which you named the sets. There is no way to continue from one set to the next. You won’t reach the end of the odds and then start the evens; your ellipses are hiding everything. Even more so in your last list: you can’t count down from ‘even infinity’ and end up at two. Think of it like a two dimensional vector (x,y). On the x axis you count by odds and on the y by evens. But they are two separate dimensions and one isn’t before the other except by convention.

              Now you can choose to put some order on the set if you like. You could say ‘I imagine that all the days are written out in 7 infinite (up and down) columns. Mondays are column 1, Tuesdays column 2, etc. This week is the zero week. So last Monday was (-1,*,*,*,*,*,*). When I imagine organizing the whole set I imagine listing out all Mondays first, then Tuesdays, etc. But of course one can’t actually list out any of it, I am just naming sets in an arbitrary order. If you are careful you might assign some ordering scheme that lets you do some algebraic manipulation, but you shouldn’t confuse it with the ‘before and after’ that apply to discussions of time. The distance between Monday -1 and Monday 0 is still seven days.

              Look back at Waters’s proof. He shows that if his numbering scheme were applicable then DF would have a finite earliest day. So far so good except for the implicit circular assumption. Then he says that this day, ‘e’ if it is not also an earliest day of D then it has a previous day (e-1) in DF. If DF is finite but you’ve made up a transfinite ordering that can be in D, this is false. D can consist of DF U some negative transfinite ‘numbers’ {w} which cannot be defined by e-1 but which are defined as w < n for all integers n. I think this is a meaningless statement when we are talking about time but it's his proof so if he wants to introduce some exotic math he needs to be much more careful with the math and with its interpretation. Perhaps he is aware of this so he adds a footnote that every day d except the possible earliest has a d-1. But that means we are just dealing with the integers and back to my original point: the integers don't have a number infinitely far from zero.

            • I’m not saying that you can just make up any old ordering on a set (such as DF) so as to claim, for example, that a statement like “this day, ‘e’ if it is not also an earliest day of D then it has a previous day (e-1) in DF” becomes false. we are dealing here with a very specific order – namely, the one determined by the way one day follows another.

              all i was doing was engaging with your original claim that “There is no such thing as a day which is infinitely distant”. i don’t know if there are infinitely distant days or not. but there are perfectly fine orderings of a countable set for which some pairs of elements have infinitely many other elements in between:

              http://en.wikipedia.org/wiki/Ordinal_arithmetic

              i’d be interested to hear you flesh out your reasons for the claim that there are no infinitely distant days.

            • josh

              “…we are dealing here with a very specific order – namely, the one determined by the way one day follows another.” That is the integers, defined by Waters by the relation that every day can be related to an earlier one by d to d-1. (Really, since days are just subdivisions of apparently continuous time he is already being sloppy and we should be talking about reals, but anyhow.)

              Everything else is exactly ‘making up any old ordering’. First, let’s not confuse distance with ordering. Pick any two integers m and n and they have a distance defined by |m-n|. Maybe you choose to indicate the integers by {m+1,m+2,….n-1; m,m-1…;…n+1,n}. So in this list m and n are separated in your order by an infinite number of elements. You can’t start at m and get to n without passing through an infinite set in between, but that is an artifact of the way you have chosen to enumerate the set.

              Now you want to move on to ordinals, but ordinals are sets, not individual numbers in the sense that a specific day is indicated by an individual label. So the ordinal ‘number’ ω represents the set of natural numbers and we can order it ‘after’ all the natural numbers in the sense that it contains them all, but it is not another number added to the natural numbers in the sense of adding +1 to any natural number. In terms of ordering you can add ω +1, +2 etc. which represents another independent set of the natural numbers, i.e. a second dimension. The wikipedia entry makes this point explicitly regarding ω + ω : “This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0′ do not have direct predecessors.” There is no way to proceed from an element in ω to one in ω+, which is what you need to talk about a succession of days. There is no defined distance between elements in the two sets.

              If you want to talk about time with this kind of bookkeeping, be aware that you are essentially introducing a second dimension of time. You can preserve an order in the sense that you have a well-defined method of bookkeeping, but you haven’t given any meaning to the idea that one dimension is ‘before’ another. The distance between elements in two different dimensions isn’t infinite, it is undefined (we could make up a metric). We might as well be talking about the set of past days and the set of shoes you could own. You don’t get from one to the other by adding or subtracting days.

              Here’s another way to think of it: imagine the graph of y = tan( x). The central region runs from x= -pi/2 to pi/2 and in this region y takes on all real values. You could think of time as the distance along this curve
              and it is infinitely long. But there is an identical curve in the adjoining region x=pi/2 to 3pi/2 (and infinite others). You can’t proceed along the central curve and get to the adjacent one. There is a distance between points on the first curve which we defined as the passage of time. This doesn’t apply to comparing points in two different curves. (You may want to say sqrt(x^2+y^2) is the distance but this would be changing the original definition of time so that it is not about a succession of points or days, and you would still find that this distance is finite for any two points.) The central curve is the set ω, the next is ω’ so ω+ω is the set of two curves considered together and ω+1 might be the set of the central curve plus one point in the adjacent curve. All the ordering information is there, but you can’t naively transliterate it into distance or succession.

            • Thanks for your clarification, Josh. I don’t think I disagree with too much of what you said (and probably nothing of real importance to the discussion of Waters’ argument). It was probably a bad idea for me to have described different orderings on the same set (the natural numbers). And yes, ordinals are sets – but they may also be thought of as order-types. Numbers are actually defined as ordinals, though – so it is not really right to say that ordinals are not numbers.

              About distance, yes the natural metric defined on the integers is indeed d(m,n)=|m-n| (the restriction of the usual metric on the reals). But different metrics may be defined. I think that in any discussion of the collection D of past days, the only metric and order that is important should have the property that, when restricted to DF, you get exactly this standard order and metric. However, if (for whatever reason) it were the case that there were infinitely distant days in D, the structure would automatically be more complicated. Is there one infinitely distant day? Are there infinitely many? Are there infinitely distant days that are infinitely distant from other days in their past? Is there an infinite sequence of days each infinitely distant from the last?

              As for the question of whether there could be infinitely distant days, I still wonder if it is possible. Your argument seems to be that because you couldn’t start at such a day and eventually get to (say) today, it doesn’t make sense (the tan graph analogy was good in illustrating your point) – please correct me if I’ve mischaracterised your argument. I’m inclined to agree this is intuitively appealing, but I’m also inclined to be skeptical of such intuitions. I feel as though I could probably construct a story in which, for example, on each of infinitely many days in the future of today, a powerful being does something that has an effect on the initial conditions of some other universe that begins at some fixed point in time. If it were possible to construct such a logically possible scenario, then it could be said that each of the infinitely many days in our universe “occurred before” the first day of the second universe.

              In any case, there certainly are philosophers who do talk about infinitely removed days in the past, and Waters does a really good job of dealing with that – he shows that you can essentially ignore any such days (if they exist), ie that they have no bearing on his argument. I think that was a good move, as it removes the possibility of coming up with an objection along those lines.

              Apologies if any of the above misses your points!

            • josh

              No, I think you’ve understood what I was trying to get across. For your parallel universe scenario you would have to be more precise in what it means to ‘have an effect on something’. There would be a correlation between events in one universe and those in another, but I still wouldn’t see any meaning in saying that one is before another. After all, it would be equally true that the events of the ‘latter’ universe predict features of the ‘former’.

              But perhaps more to the point, if we assume that your parallel universe should be described as you posit, then it doesn’t fit into Waters framework. D would be infinite and DF would be finite. But you can’t reach D from DF because e-1 is not a day in D-DF (taking e to be the first day in DF). Waters’s construction makes D the same as DF.

              I want to emphasize that this is a side issue and the central problem with Waters argument is much more trivial. I just think he is guilty of adding on a lot of seeming sophistication to cover up the bad heart of his point, and a little knowledge shows that the ‘sophisticated’ bits are sophomoric.

              About numbers being ordinals: I glossed over some details. Cardinal and ordinal numbers are generalizations of the natural numbers. In the ordinal number system, ordinal numbers equivalent to the natural numbers exist. So ‘2’ represents the set {0,1} and 1 represents the set {0}, etc. Although it shares many properties this isn’t exactly what we mean by assigning a number to a day. Wednesday does not represent the set {Monday, Tuesday}. With the order type definition, 2 is a symbol describing all sets that share an order-preserving bijection with {0,1}. Again, you wouldn’t use it in the same sense to label an individual day.
              Suppose I have an infinite number of pairs of shoes. I label the left shoes 0,1,2… and the right shoes ω,ω+1… That’s labelling, I could as easily have used 0,1,2 and 0′,1′,2’… or (1,1),(1,2)… and (2,1)(2,2)…
              The order type of either left or right shoes is also denoted ω and the collection of pairs by 2ω. ω, the first right shoe, is not the same as the order type ω which tells me that the collection of left shoes is infinite in a certain way. I blame Cantor.

            • dammit, i just had a big comment deleted…

              i can’t remember most of what i said, but i was thinking about the possibility of DF being finite while D is infinite. this would necessitate the ordering on the set D to be w+n for some n (or possibly with something more complicated before the “+n”). now this does seem strange, but consider the following possibility:

              there is a universe that will go on forever – infinitely many days in the future. the universe was created by some being. it creates beings who live in the universe and constructs a test – those who pass the test will, when they die, end up in another world called “heaven” – those who fail the test end up in another world called “hell” – every day, a finite but nonzero number of people die who fail the test, same for those who pass the test. heaven and hell have a starting day, and on this day, each person who has lived in the original world will be in either heaven or hell.

              that might need to be fleshed out a bit. but i think it paints a reasonable scenario in which it could be said that the original universe “happened before” heaven and hell. (thinking about it now, there’s probably no need for there to be *two* additional worlds.) and, it also couldn’t be said that the first day of heaven/hell has an immediate predecessor in the original universe, so there are infinitely distant past days, no?

              i agree that these are essentially side issues (but interesting ones), though now i’m wondering if the [D infinite] => [DF infinite] implication is shaky, and not such a minor problem. i’ll sleep on it…

              it’s good to talk this stuff over with someone who knows their math, btw.

            • Waters doesn’t enumerate his assumptions about D, but I believe these are the assumptions implied:

              1. D is totally ordered.
              2. D has a maximum element, which we call t-1.
              3. If d is in D, and if the set S = { x | x is in D; x > d} has at least n elements, then there exists an nth smallest element of S, which we define as d+n.
              4. If d is in D, and if the set S = { x | x is in D; x < d} has at least
              n elements, then there exists an nth largest element of S, which we
              define as d-n.

              Nowhere does Waters assume that there is a "distance" defined between any two days in D. Josh, do you believe that this can be proven from my assumptions, do you believe that I've misinterpreted Waters' assumptions, or do you believe that it's something Waters *should* have assumed?

            • that’s a nice list. i haven’t thought about whether it is complete. but it makes me realise that there could even be a problem with how d+n is defined:

              “for any d in D and any positive integer n such that d precedes t by at least (n+1) days let (d+n) represent the unique nth day following d in D.”

              in particular, suppose there are infinitely distant days in D, but there is a latest such day, say day e, and suppose DF is infinite. then day e is before all the days of DF, so i guess we’d say day e precedes t by at least (say) 2 days. but then we’d need to be able to define day e+1, which doesn’t seem to exist.

              or does “precede” need to be understood differently? i take “d precedes t by at least (n+1) days” to mean that there are at least n days c such that d<c<t which would make the previous paragraph seem a bit troublesome.

              i wonder if it is necessary to do away with infinitely distant days, i.e. assume they don’t exist in order to make these problems go away. i don’t think this would be a terrible thing to do. if an argument for the finitude of the past could only be attacked by speculating about properties of hypothetical infinitely distant days, i think it’s doing pretty well. (but of course there are more basic objections to waters’ argument.)

            • no idea what all that =” ” stuff is all about. i’ve tried to fix it a few times and given up now…

              but on further thought, it probably suffices to define d+n by just saying “if there is a day n days after d, we define that day to be d+n”. i think that solves the problem.

            • josh

              I was just browsing through some old posts and saw this reply so I figured I’d respond, albeit 24 day late. 🙂 I’m looking at what Waters wrote: “Finally, let DF represent the sub-collection of all days in D that are finitely distant in the past 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 past.”

              I don’t see how Waters could not be talking about a distance here. Also, note that Waters includes the assumption that all elements in D are related by d-1 existing for all d except possibly the earliest. Which maps them to the integers which have a defined distance.

              I think in your list of 4 assumptions you didn’t actually mean that d+n is the nth ‘smallest’ element since that is only true for d being the earliest. Aside from that, it looks like in your list, every other element in D can be related to some d by d+n or d-n. Which is to say that they are the integers or some subset and there are a finite number of elements between any two d+n and d-m. On the other hand, the wording ‘S has at least n elements’ complicates things. If I allow that some S is the set {d}+{the set of whole numbers} so that d should be labelled ω perhaps, then it’s not true that I can label the other members by d-n even though it clearly contains at least n elements. But this doesn’t fit in with Waters’s description. (and would explicitly allow an infinite past by his conventions).

            • Josh,

              “I don’t see how Waters could not be talking about a distance here.”

              I think Waters is just being imprecise. For some pairs of days, a distance can be defined. For other pairs of days, no distance is defined, or as Waters puts it, they are “infinitely distant”.

              “Also, note that Waters includes the assumption that all elements in D are related by d-1 existing for all d except possibly the earliest.”

              He does not include this assumption. Every day d (except the earliest) has a d-1 defined, but that does not mean that you can use the -1 operation to get to every day. His assumptions are a bit like the Peano axioms, but he’s omitted the axiom of induction.

              “I think in your list of 4 assumptions you didn’t actually mean that d+n is the nth ‘smallest’ element since that is only true for d being the
              earliest.”

              I said d+n is the nth smallest element of S, not of D.

              “Aside from that, it looks like in your list, every other element in D
              can be related to some d by d+n or d-n. Which is to say that they are
              the integers or some subset and there are a finite number of elements between any two d+n and d-m.”

              But this is not true of the example of D I offered in my first reply to you. Do you think my example breaks any of my axioms?

              “On the other hand, the wording ‘S has at least n elements’ complicates things. If I allow that some S is the set {d}+{the set of whole numbers} so that d should be labelled ω perhaps, then it’s not true that I can label the other members by d-n even though it clearly contains at least n elements.”
              I don’t understand what is being said here. I don’t know how you’re defining S.

            • couldn’t a distance be defined between any two days d1 and d2 by declaring the distance to be 0 if d1=d2, or else n+1 where n is the number of days d such that d1 < d < d2? in this case, the distance is defined even if the days are separated by infinitely many days. it would be a different kind of distance function, and not completely illuminating in some cases – eg, knowing that two days are infinitely distant does not tell you about the structure of the interval between them. i suppose you could define the "distance" to be the order type of the interval between the two days – this would encode a lot of useful information.

            • josh

              ‘For other pairs of days, no distance is defined, or as Waters puts it, they are “infinitely distant”.’

              Well, ‘infinitely distant’ and no distance defined aren’t the same thing. But again, look at how Waters is trying to reason: “Now, if e is not also an earliest day of D then it has a previous day (e-1) in DF, which contradicts the fact that e is the earliest day of DF. ” So if you want to allow that e-1 is somehow defined in an ordered way but it has an undefined distance from e which Waters hasn’t specified, it follows that e-1 is not in DF, the set of finitely distant days, no contradiction.

              Now,
              if e is not also an earliest day of D then it has a previous day (e-1)
              in DF, which contradicts the fact that e is the earliest day of DF. –

              “He does not include this assumption. Every day d (except the earliest)
              has a d-1 defined, but that does not mean that you can use the -1
              operation to get to every day.”

              So here, you are worried about multiple disconnected infinite sets, or loops? I think we can leave out loops for now given his talk about unique d-n’s and order relations. I didn’t read him this way, but you are right that he could be construed as allowing for two or more infinite sets such that all d’s have a d-1 but you can’t get from one set to the other. I see that in your original reply to me this is what you were aiming at. Let’s call such a set(s) D1+D2 where D1 < D2 in our ordering. Then we get the result that if D2 is finite e-1 must be in D1 but if it is infinite then you can't get to D1. This is due entirely to his footnote, which isn't in your enumerated assumptions as far as I can see. Otherwise we could have my scenario where D1 is infinite and D2 is not. I.e. it isn't true that the 'earliest' day has to be the only one with no d-1, if we're allowing these disconnected sets.

              But I agree, before assuming that DF is finite, D, as stated, could contain days with no defined distance from t but infinitely many elements in between in the ordered list. Really his footnote amounts to: 'By assumption, either D is a finite set with connected days, or it is infinite and may then contain multiple disconnected infinite series, only the earliest of which may have a first element.' I maintain that is a weird way to define 'the collection of all days prior to t'. What he wants is the reasonable 'assumption' that talking about the past means we can label the days with integers going backwards from t. Which would be fine but on your reading he goes to this extra trouble to allow for d's that aren't part of that set only to rule them out with the assumption that if his integer labelled days are finite they constitute the whole set.

              "I said d+n is the nth smallest element of S, not of D."

              My mistake, apparently I misread.

              "I don't understand what is being said here. I don't know how you're defining S."

              Same S as you. Take your example with the set {e,e+1…} + {t-m} where d is in {t-m}, and let t-m terminate at some finite m. Then there are more than m earlier days x< d but I can't label the ones beyond t-m with some d-n.

            • Josh, I think we have an understanding now. Yeah, it seemed to me that Waters was allowing for multiple disconnected sets, thus the distinction between D and DF.

              Personally I wouldn’t worry about multiple disconnected sets, since it seems like such a bizarre picture of the past. I would have taken a stronger set of axioms where D=DF. But I suppose Waters should take the weakest set of axioms possible such that his theorem still holds.

    • Ben Waters

      Hey James,

      I thought I’d share with your readers a condensed version of my reaction, which I’ve already shared with you privately at greater length via e-mail, to these two very stimulating critiques of my argument that you’ve written. I hope you don’t mind.

      First of all, you claim here that my argument tacitly assumes what you are calling Assumption A, but this isn’t true. Rather, the fact that Methuselah will never write about future days follows from what I explicitly stipulate, not assume (implicitly or otherwise). My understanding is that you now retract this point, so I won’t pursue it any further.

      Secondly, you claim here that my argument is circular. The problem seems to be that (2) relies on a sub-claim to the effect that my Methuselah’s diary scenario is logically possible, but that is true only if D is finite and whether D is finite or not is precisely the point at issue. I don’t think this line of criticism works because it would entail that other perfectly valid arguments are circular as well. For example, the proof of Gödel’s first incompleteness theorem, which says (following Kleene 1967) that a broad class of formal systems can’t be both consistent and complete, relies on a sub-claim to the effect that within any such formal system there exists a proposition that is true if and only if it is not provable, but, of course, this sub-claim is true only if Gödel’s first incompleteness theorem is true. So, according to this logic, the proof of Gödel’s first incompleteness theorem would have to be considered circular as well, which can’t be right.

      That said, it may still be the case that my argument is (in some sense) circular if I have no good reason to think that my Methuselah’s diary scenario is logically possible and simply assume that it is. However, as you know, I do think there are good reasons for thinking that my Methuselah’s diary scenario is logically possible. In particular, I think the Methuselah’s diary scenario seems logically possible on the basis that Methuselah’s powers of memory and dispositions concerning his diary, although somewhat idealized in the former case and artificial in the latter case, are very much like the sorts of powers and dispositions we can have in the actual world, which suggests that the diary-keeping activities entailed by Methuselah’s powers and dispositions are eminently performable (hence also logically possible). Indeed, I suspect this line of reasoning cannot be denied without coming across as being overly skeptical. In any case, I chose not to explicitly record this last bit of reasoning in my article because I thought it should be obvious, but maybe that was a mistake on my part. In sum, I don’t think what you’ve said here gives us any good reason to think that my argument is in any sense circular.

      • I don’t mind at all, Ben – thanks for sharing your responses. I’ve enjoyed our correspondence and I think my readers will also! When I’ve written my response to your response more formally, perhaps I’ll post that also.