William Lane Craig Tristram Shandy Paradox

William Lane Craig

William Lane Craig's Version

The original version of Tristram Shandy is used by Bertrand Russel and essentially is as follows: "Tristram Shandy is a character who is writing his own autobiography. He writes so slowly it takes him an entire year to record the events of a single day. It becomes apparent that he will quickly fall behind in his progress. Therefore he will never be able to complete his work. Russell then claims; “if he had lived forever, and not wearied of his task, then, even if his life continued as eventfully as it began, no part of his autobiography would have remained unwritten.” seeBertrand Russell's Tristram Shandy Paradox for details.


William Lane Craig sets out to demonstrate that the principle of correspondence and an actual infinity only work in the mathematical realm. According to Craig it is when the principle of correspondence is applied to an actual infinite in reality that various absurdities arise. An actual infinite is a complete infinite series opposed to the potential infinite which only increases indefinitely remaining at any time a finite collection. His motivation is to demonstrate that an actual infinity cannot exist in the physical world as opposed to the mathematical realm, and therefore when we envision an actual infinity in the world and apply the principle of correspondence the results are paradoxical.



Craig alters the Tristram Shandy Paradox to refer to the past to bring out the apparent paradoxes that arise with the use of the principle of correspondence. Imagine that Tristram Shandy has been writing for any eternity past and that the past is an actual temporal infinite. Craig claims that a one-to-one correspondence can be established between days written about and years of writing. According to Cantor this means that the set of days recorded and the set of years of writing are equivalent sets. Therefore, for every year of writing there is a day recorded, and every day lived through has a year in which it was recorded. For Craig this entails that given an infinite amount of time Tristram will have finished writing his autobiography. Craig seems to be led to this conclusion based on the fact that if both set have an equal infinite amount of members this entails all the days in the past have been recorded. That in fact Tristram would be finished writing which would be at the same time impossible. Craig states this is clearly absurd since “Tristram Shandy could not yet have written today’s events down.” (1, pp33), for today would require at least the next year, if not some year somewhere in the future, to be recorded. In fact, we are aware Tristram would never finish, “for every day of writing generates another year of work.” (1, pp.33), and poor Tristram falls progressively behind. If the principle of correspondence really related to the real world as opposed to the mathematical realm, then Tristram would be finished. Therefore, if we are to accept the existence of an actual temporal infinity of the past and the principle of correspondence we are led, it seems, to Tristram both finishing his autobiography and also falling behind.


If the theory of correspondence entails that Tristram would be finished with his book, then Tristram could have been finished at any time in the past. Craig states this is because yesterday and the day before, and so on, also have an infinite past. If in an infinite amount of time he could have completed his work by the present, then since every past day has an infinite pasty he could have finished at any point in the past. But if Tristram could be finished at any point in the past then anytime in the past he will not be writing, however, we assumed him to be writing for all eternity. Thus, “at no time in eternity will we find Tristram to be 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.” (1, p.34). What makes the situation absurd is not the fact that he will never finish but the claim the correspondence leads to him being finished and therefore not writing at all. The question then becomes whether the principle leads to the conclusion that Tristram at any time could finish writing.


One reason Craig believes Tristram should be finished writing by the present is that he has an infinite amount of time to accomplish the task. Craig states Bertrand Russell gives this reason; “in order to write about an actually infinite number of days at the rate of one day per year all one needs is an actually infinite number of years.” (1, p.100). Craig states that because there is a year for every day recorded in the past there ought to be plenty of time available to write about all the days.


To complicate the issue Craig claims Tristram will have fallen infinitely far behind. Craig states “At any point in the past or present, therefore, Tristram Shandy has recorded a beginning less, infinite series of consecutive days. But now the question inevitably arises: which days are these? Where in the temporal series of events are these days recorded by Tristram Shandy at any given point?” (1, p101). He concludes these days must be infinitely distant from the present. If this is true then the interval between the last day recorded and the present is infinite. He states this interval must be infinite because if Tristram has been writing for an infinite amount of days the distance will be infinite. In Russell’s version of the scenario, occurring in the future as a potential infinite, the gap between the last recorded day and his current year of writing progressively gets larger as time passes, although it always remains a finite interval. With an actual infinite in the past Tristram has already been writing for an actual infinite amount of time and Craig reason the gap between the last recorded day and the present will be infinite. At no time would the gap between the last recorded day and the present have been finite. For at every moment in the past Tristram has been writing for an infinite amount of time, and therefore Tristram has always been infinitely far behind in his writing.


If Tristram is infinitely far behind this implies that an infinite past entails infinitely distant events. Craig’s point is that no matter how far we mentally regress we cannot find the last day written. Even though we cannot locate the day we can still create a correspondence between the set of recorded days and the set of years of writing, as both have the same cardinal number. This brings up the medieval problem of traversing the infinite. It appears impossible that Tristram could write an infinite amount of days, and thus traversing an infinite interval, to reach the present. Likewise it seems impossible for events that were once present to recede to an infinite temporal distance. The once present event would have to recede through an endless amount of days to somehow get beyond infinity.


There is no problem with the statement that given an infinite amount of days an equally infinite amount of days can be recorded, especially if it takes Tristram a day to write about each day. However, given an infinite amount of years this does not necessarily entail that the infinite set of days recorded includes all the days in the past up to yesterday (yesterday being the last completed day). Yet, Craig states “There is no day of the past to which a past year does not correspond, therefore, assuming that his goal is to record every day of his life, Tristram Shandy should finish the job in an infinite number of years, which by now, he has had).” (1, p.100). It is a valid statement that every year of writing has a day recorded corresponded to it, however, it is not true that every year of writing is corresponded to every day lived. For Tristram to be finished he would have to have recorded yesterday, but yesterday requires a future year of writing to be recorded. Therefore yesterday and many other past days lived through do not have a year of writing to correspond to, those being future years.


The first issue we must analyze in depth is whether the principle of correspondence leads to the conclusion Tristram would be finished writing at any point. If he could be finished writing then we have the paradox of him both being finished writing and at the same time so far behind that he will never finish. In addition we have the issue of whether there is a finite or infinite interval between the last day recorded and the present. If we decide Tristram is infinitely far behind we have the additional problem of how he could traverse an infinite interval to reach the present and how a once present event could recede beyond infinity. Phrased this way we have a clear paradox for Tristram can be described as both being far behind and finished at any point in the past.


Quentin Smith responds


Smith takes issue with the claim the principle of correspondence leads to Tristram finishing his work. He states it is false to assume the number of past days written about is the same as the number of number of past days means there are no past days unwritten about. He brings up the concepts of a set and its proper sub-set. A proper subset is a subset that does not include all the members of the original set; therefore there is at least one or more member(s) of the original set that do not belong to the proper subset. According to Smith the correspondence principle does not entail that Tristram will ever finish writing. The number of past days recorded is a proper subset of the infinite number of past days, and a “proper subset of an infinite set can be numerically equivalent to the set even though there are members of the set that are not members of the proper subset.” (#3, pg 87). Just as the “ infinite set of natural numbers has the same number of members as its proper subset of even numbers, yet has members that are not members of this proper subset (these members being the odd numbers); so the infinite set of past days Tristram has lived through has the same number of members as its proper subset of days written about, yet has members that are not yet members of this proper subset (these members being the days unwritten about).” (#4, pg 87-88)


Ellery Eells Responds


Ellery Eells argues against William Craig by taking issue not with his use of the correspondence principle but with the claim Tristram would be infinitely far behind. He argues the interval between the last day recorded and the present year of writing will be finite. Eells notes, there “will always be only a finite number of days between any given day and the day on which he finishes describing the given day.” (#5 p.454). One reason for this is simply that if we know the last day recorded and the series ends at the present we have constructed a finite interval with two ends. Since the series is consecutive if the interval has two definite ends the interval will be finite. Eell’s states that we could find the day Tristram began writing about the past because the present must be finitely distant from the last day recorded. We are aware that the series of days recorded and the series of years of writing diverge more and more as time passes. Likewise if we regress member by member the gap between the day recorded and the year that Tristram is writing about that day shrinks. As we regress more years we will eventually find the day that Tristram began writing about the day he was then living. However, if we supposed Tristram to be writing for all eternity then there should not be a beginning to his writing.


Eells goes farther by stating Tristram could not have always been writing about his past, so before that he must have been writing about his future. If we want to adhere to the premise that he has been writing for all eternity, then we must assume that prior to the first day he began writing about his past he must have been writing about his future. Furthermore, only after infinitely many years writing about his future could he then begin to write about his past.


It does seem by taking the original Tristram illustration and flipping it into the past as Craig has done we have a version that presupposes a beginning to his writing. It does not appear there would be infinitely distant events in a series that is consecutively ordered and composed successively of discrete members (each member of the series would be understood as having a finite duration). As the series is composed of finite events with no gaps, then between any two points, there would be a finite interval only and we would find a beginning to Tristram’s writing.


Robin Small Responds


Robin Small takes issue with the claim a one-to-one correspondence can be established in William Craig’s version of Tristram Shandy. The original version of Tristram Shandy used by Bertrand Russell establishing a correspondence is not a problem since both sets have determined first members as the first day of Tristram’s life corresponds to the first year of his writing. Small states that Craig uses the principle of correspondence illustrate Tristram would be finished writing and since this is clearly impossible we must reject an infinite past. This is assuming a correspondence can be established. When Craig turns the illustration into the past, thus making it a paradox, there are no first members to the set of years of writing and the sub-set of days recorded. When we try establishing a correspondence from the last members of each set, we cannot locate the last recorded day to match up with the present year of writing. This is blatantly apparent if there are infinitely distant events as Craig claims; how can you match the last recorded day to the present year of writing if you cannot even locate that day? Therefore Small concludes a correspondence cannot be established to begin with.


Nevertheless we are aware that both sets are denumerable infinites and therefore equivalent. It is Small’s claim that since we cannot establish a physical correspondence, as we cannot locate the last terms and there are no first terms, then we cannot establish a correspondence at all.


Conclusion


William Craig claims the principle of correspondence leads to Tristram having finished his autobiography by the present. However, since the duration of one term of the set of years of writing is 365 times the duration of one term of the set of recorded days, as time progresses the sets diverge more and more and will not be aligned by the present. Secondly, the set of days recorded is the proper sub-set of the set of days Tristram has lived through. As a result there are days that Tristram has lived through that are part of the entire set, but are not included in the proper sub-set. For Tristram to be considered finished he would have had to record every part of his life, when in fact we find the proper subset does not include every part, as all the unrecorded days are excluded. Thus the sets remain equivalent without entailing Tristram is finished his work and certainly will not be finished by the present. In addition to the fact that not every part of the set of days will be recorded we find that its subset of days recorded is not as extensive as the set of years spent writing and therefore Tristram is behind in his work.


If Tristram is only finitely far behind in his work, and we can locate the last recorded day, then by regressing a year for every day we would be able to determine the day on which Tristram began writing. This directly conflicts with the premise he has been writing for all eternity. Therefore he must either be infinitely far behind or the scenario itself presupposes a beginning. According to Craig, if Tristram has been writing for an infinite amount of time then likewise the interval between the last day recorded and the present will be infinite. This being the case it seems impossible for a once present event to recede day by day to an infinitely distant location from the present. If it is impossible for Tristram to be infinitely far behind and being finitely far behind determines a starting point, as Craig concludes, an infinite temporal regress is impossible. If we are talking about a finite collection becoming infinite by successive elapsing of days the response might be in an infinite amount of time an infinite amount of days can elapse, and then of course more from there. However it does seem infinitely distant events in a consecutive continuous series of days is possible in this case. It is not impossible for an actual infinite amount of days to exist. No matter how many years Tristram writes he does not write more than a finite amount from the event that was once present, for the potential infinite is only a finite collection that increases indefinitely. Therefore he would not be infinitely far behind as there would not be infinitely distant events in the series. If there are not infinitely distant events then we can find when Tristram began writing. It is the illustration itself that presupposes a beginning and there is no paradox of having infinitely distant events.


In fact, Robin Small states we can create an illustration properly turned into the past that shows no paradoxes. In Russell’s version it takes a year for Tristram to write about one past day and the series has a first term but no last term. In Craig’s version it also takes Tristram a year to write about a past day but the series has no first term and has a last term. Craig only alters where the beginning and end are. So Small gives us a story that would really turn the paradox about. His alternative goes thus: “suppose that Tristram Shandy plans the events of each day in his life in advance, and it takes him a year to do this for each day. Could he have completed his task up to the present day? If he had been carrying it out through an infinite past, the answer seems to be affirmative. For today’s events were planned in a certain year, yesterday’s events were planned in a previous year, and so on.” (2, p. 215). In Russell’s version, being as it was a potential infinite going into the future, we could always find a year further in the future to record a further day. Similarly in Small’s version we can always find a year further in the past that planned a given day. Again the set of years is equivalent to the set of planned days and are in one-to-one correspondence. There is also no paradox from the conclusion Tristram has finished this task up to the present. In addition we have a set that is infinite but finite between any two points and therefore there are no infinitely distant events, for the year that planned today was last year. Although the gap between the years of planning and the day planned increases the farther we recede into the past, the interval remains finite. Similarly in Russell’s version the gap between the last recorded day and the year of writing increases as we progress into the future it remains finite as well. Even though the series is finite between points we do not have Eell’s dilemma for we never locate the beginning of Tristram’s planning.



Notes


#1 Craig, William Lane; Theism, Atheism, and Big Bang Cosmology; Clarendon Press, Oxfordm NY1993


#2 Small, Robin; “Tristram Shandy’s Last Page” British Journal for the Philosophy of Science, V37 1986


#3 Smith; Theism, Atheism and Big Bang Cosmology; Clarendon Press, Oxfordm NY1993


#4 Smith, Quentin; Kalam Cosmology;


#5 Eells, Ellery; “Discussion: Quentin Smith on Infinity and the Past” Philosophy of Science V55 1988


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