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):
- Infinite dreams
- Infinity minus infinity
- Can God count to infinity?
- Arguing about mathematical objects
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.
