- Education and Science
Bertrand Russell's Tristram Shandy Paradox
In the Principles of Mathematics Russell brings up the original Tristram Shandy scenario. This illustration is used to figure out if the Cantorian principle of correspondence creates any contradictions as the axiom ‘the whole is greater than the part’ does when applied to the infinite. See Theory of Coresspondence vs The Whole is Greater than the Part
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.” (Russell, p. 358). The set of days recorded and the set of years spent writing each have a first term with no end. As each series has a first term it is easy to pair years to days in a correspondence. Thus every day will have a year of writing matched to it, and every day will, eventually, be written about. There being a one-to-one correspondence between terms means the sub-set of days recorded is equal to the set of years of writing. Russell suggests this means the whole is similar to the part. Russell concludes the Cantorian principle is valid for infinite sets, since unlike the axiom the whole is greater than the part, no contradictions arise from its use.
There is no problem with the statement that each day recorded can be correlated to a different year of writing. Both sets are increasing in members at a ratio of 1:1 and therefore will always be equivalent. This does not entail that Tristram will be finished his writing or that he will be caught up at any moment, it only demonstrates that both infinite sets are the same size and from the perspective of being instantiated it time are increasing at a determined rate. Because we are looking at a series with a first term and no end term we are talking about a potential infinite and therefore there will always be a further year of writing in the future to record a past day. As time passes, the further we progress into the future the more days become recorded. Since we can potentially progress into the future indefinitely there is no past day in the series that will remain unrecorded. Russell concludes; “the Tristram Shandy proves that two variables which start from a common term, and proceed in the same direction, but diverge more and more, may yet determine the same limiting class.” The set of years of writing is more temporally extensive than the set of days recorded; even though the sets diverge more and more they are equivalent sets that are indefinitely increasing at the respective rates of one day per one year. As Russell claims, the correspondence between terms does not entail any absurdities.
It is also true that given any point in the series the correspondence holds but we clearly see that Tristram is behind and not caught up in his writing. There will be many days Tristram has lived through but are not yet members of the subset of recorded days. Obviously these unrecorded days will correlate to future years of writing. Therefore the correspondence that allows us to say both infinite sets are the same size, does not entail they are matched in duration or that during any time we choose in the series he will be caught up. Looking at the set from within a finite portion of the series enables us to see how it is instantiated in time and this tells us that at any point Tristram will not be finished writing. We realize he will never finish writing. Therefore a correspondence in this case does not entail he will finish writing.
Russell’s conclusion that the whole is similar to the part may be misleading if we think ‘whole’ is necessarily a finite term and entails a series with two definite ends. However, Russell makes a distinction in the definition of ‘whole’. He distinguishes “wholes which are defined extensionally, I.e. by enumerating their terms, from such that are defined intentionally, I.e. as the class of terms having some given relation to some given term, or, more simply, as a class of terms.” (Russell, p.349). The whole that is defined extensionally is necessarily finite. A “finite whole may be taken collectively, as such and such individuals A, B, C, D, E say. A part of this whole may be obtained by enumerating some, but not all, of the terms composing the whole; and in this way a single individual is part of the whole.” (Russell, p. 360). Finite wholes and parts may also be defined by intension. Thus “we know without enumeration that Englishmen are part of Europeans; for whoever is an Englishman is a European, but nice vice versa. A whole defined intensionally does not require enumeration and can be an infinite collection that refers to ‘a whole whose parts are the terms of a class’” (Russell, p. 349) A whole taken intensionally then does not refer to a collection with definite limits but to a class which definite items either belong to or not. It is as such defined without reference to any specific individual and without enumeration and thus a whole in this sense can be infinite. If “a be a class-concept, and individual of a is a term having to a that specific relation which we call a class-relation.” (Russell, p.360). If “now b be another class such that, for all values of x, ‘x is an a’ implies ‘x is a b’ then the extension of a (ie the variable of x) is said to be part of the extension of b.” (Russell, p.360). Here there is no enumeration of individuals needed. To say a and b are similar is to say “that there exists some one-one relation R fulfilling the following conditions: if x be an a, there is a term y of the class b such that xRy; if y be a b, there is a term x’ of the class a such that x’Ry’” (Russell, p. 361). Russell gives the example of prime numbers in which primes are the proper part of integers, but this cannot be established by enumeration. It is deduced from “if x be a prime, x is a number’ and ‘if x be a number, it does not follow that x is a prime.’” (Russell, p.361). That the class of primes “should be similar to the class of numbers only seems impossible because we imagine whole and part defined by enumeration.” (Russell, p.361). Regardless, we are able to say the infinite subset correlates equally to the infinite set.
Seeing the infinite as a whole or complete is not a universally accepted notion. Some people see the infinite as progressing toward a limit that it never reaches, for every member the set receives there is always more to take. As such, they accept the validity of only the potential infinite. Any series or set is seen as able to continually acquire more members without becoming complete, and therefore increases indefinitely. According to Cantor the potential infinite is considered the improper infinite. An actual infinite is considered to be the true infinite then; such as the totality of numbers as a completed unity. Opposed to, say, trying to count all the numbers we step away and grasp them conceptually as a whole. The series of whole numbers would be considered a complete infinite series since the series contains every whole number, even though, obviously we cannot count every whole number in the series.
Russell’s use of the Tristram Shandy example does not create too many issues if taken as a potentially infinite series. He does not state Tristram will be finished and obviously at any point in time he is not finished. At the same time we can clearly state that no part pf his autobiography will remain unwritten as time progresses in this potentially infinite series. The use of the correspondence theory principle does not, then, entail Tristram will complete his work. It does entail that the whole is similar to the part in infinite collections.
Although by using a potential infinite and having the example instantiated in time can create some perplexing issues. For example we are not looking at the series from within the series, for within the series only a finite amount of time has passed from when he began writing to the current day and clearly at any present moment he is vastly behind in his work. It is more of a problem to be looking from outside of the series. We have specified Tristram lives forever, and assuming the universe and time continues forever, and he does not get bored of his task, he will have been writing for an infinite amount of time. If the infinite is conceived as a collection of indefinitely increasing but remaining at any point in time a finite collection the problem of traversing the infinite becomes a puzzle. At what point does he go from writing for a finite amount of time to having been writing for an infinite amount of time, when there is always another day to add and the duration between the beginning and any day is finite? Just like if I started counting days and lived forever I would never count to infinity. This is an issue with a scenario where we have a beginning term like this. While it is potentially infinite at any time it is a finite collection that just adds terms indefinitely… never reaching the infinite.
Setting aside that issue others have a problem with the infinite collection in this case being seen as a whole. If he had been writing for an infinite time then surely he would have the time to have written about an infinite amount of days, therefore at some point he would in fact be finished writing. However, just because the set of years of writing it equivalent to the sub-set of days recorded in terms does not mean there is also not a large sub set of days not yet recorded. There being a distinction between the number of terms and their duration and extensiveness when instantiated in time. Therefore, odd as it sounds even if he wrote for infinity and recorded an infinite amount of days he still would not be caught up… and who knows maybe he would fall infinitely behind.
P.O. Johnson's Response
P.O. Johnson claims that the principle of correspondence does not aid us in clearing up any paradoxes of the infinite and may indeed cloud the issue further. He points out that by applying a correspondence we are not able to state that one series is more extensive than the sub-series. Johnson makes the valid point in saying that since there is always a further member to pair up hides the fact “that the sub-series is using up its terms at a much faster rate,” (Johnson, p.369). Here Johnson is referring to the series of whole numbers and the sub-series of even whole numbers. Indeed, if the series is stopped at any point, and thus made a finite whole, then we could easily establish the whole is greater than the part. Johnson states if we halt the series of whole numbers and we count backwards “it is obvious that the even number sequence will reach zero in the fewest terms.” (Johnson, p.369).
This can be applied to the Tristram Shandy Paradox in which the set of years of writing is far more temporally extensive than the set of recorded days. For every day Tristram records of his past there are 365 days of writing. This is important to note how the infinite series is said to be instantiated in reality and time. For instance, since the set of years of writing is more extensive than the set of recorded days a greater duration of time is traversed in the set of years opposed to the set of days. Thus in this scenario we are aware that the members of the set of years spent writing and the members of days recorded diverge more and more and will never match up in time. If we ignore this fact then we may be let to believe because the sets are equivalent the principle of correspondence leads to Tristram Shandy finishing his autobiography, when clearly that would not be the case. The set of days Tristram has lived through (of which the set of days recorded would be a sub-set to) is as extensive as the set of years spent writing and those two sets are also equivalent through the principle of correspondence.
The Cantorian principle merely reflects a way to compare sizes of the infinite. Johnson suggests that Bertrand Russell makes a stronger claim by stating the principle entails the whole and part can be similar, as expressed in the Tristram description, where the set up days recorded and set of years of writing are similar.
“There is no paradox in the fact that Tristram could not complete his autobiography at the rate he is going, even if he lived forever; it is only when one comes to consider that, on the other hand, if he were to live forever, he would apparently be able to complete every part of it, that one arrives at the paradox, which is that it seems to be true that he can never finish his autobiography and that no part of it need remain unwritten.” (Johnson, p.371)
Clearly Johnson sees no problem with Tristram not finishing his work, due to the fact the set is far more extensive than the subset. However, Johnson seems to be implying that because the whole is equal to its parts that Tristram could be interpreted as finishing ‘all’ the parts of his autobiography, therefore finishing, that he finds to be the contradiction. Thus since correspondence entails the whole and part are equal we are led to the paradoxical conclusion that Tristram is both finished writing and never finished writing. Yet just because the sets are equivalent does not entail Tristram will be finished writing. Due to that very extensiveness in time he notes this is not the case, however due to the nature of the infinite in this case it is also true there is always another year of writing and always another day recorded. No day in all the days Tristram has lived will remain unrecorded. Even if they are both infinite sets and equivalent they can still be added to by another year of writing and another day recorded. Johnson fails to notice that while the set of years is equivalent to the set of days recorded, the set of days recorded is only a part of the set of days Tristram has lived through. The set of all the days Tristram has lived through also contains all the unrecorded days of his life. Both the set of days Tristram has lived through and the sub-set of days recorded are equivalent to the set of years spent writing. Only the set of days Tristram has lived through is as temporally extensive as the set of years spent writing. Therefore, the subset of days recorded is only a part of the set of all the days, and ‘every part’ of his autobiography is not recorded. For despite the correspondence found with the set of days recorded and the set of years spent writing there remains another ‘part’ of the set of all the days Tristram has lived through, that subset consisting of all the unrecorded days.
Just because Tristram writes for an infinite amount of time and, say, records an infinite amount of days, and those sets are equivalent, does not imply he would be finished. Correspondence seems to cloud the issue when in fact it is not implying anything beyond the fact the infinite subset is equivalent to the infinite set. If we instantiate it in reality there can be some differences noted in extensiveness and duration of sets vs. subsets that help clarify this.
Johnson points out that once we discuss the infinite, notions of wholes and parts becomes inadmissible. And he further notes that the series cannot be whole for it is not completed, thus there is no finished work. The intuition that an infinite is not completed is a common one. Indeed from within the series the infinite is endless. It is pretty endless if I began counting all the whole numbers. However, we can have a whole infinite series when we are looking at the series from beyond itself. The series is said to be complete because it holds all the whole numbers. Perhaps we might want to note that in Russell’s Tristram Shandy series we do not have a completed infinite series, we have a potentially infinite one. As such no matter what year Tristram is currently writing in there is still a finite span of time from that point to the beginning. Furthermore Johnson states, it is true that if Tristram lived forever he would write about any day, but not ‘all’ days or ‘every’ day.
Johnson casts doubt on whether a one-to-one correspondence is valid in this situation. The days and years do not run in a parallel series and the discrepancy between the series increases each year. At any point there are many days Tristram has lived through that have no corresponding term. Which is entirely true if we look at the series at any given point and not as a whole infinite series, a problem that arises with looking at a potential infinite. Those unrecorded days are not yet part of the ‘days recorded’ subset, yet they will also not remain unrecorded. A correspondence can be established and as each set is a denumerable infinite we know them to be equivalent, but by stating they are equivalent we can ignore the fact that, in time, one is far more extensive than the other. So we know even though we have, if we can say this is a true infinite, equivalent series’ correspondence does not entail Tristram is at any time complete. Since, it is also apparent that the sub-set of unrecorded days is also equivalent.
William Lane Craig's Response
William Lane Craig implies it is in fact Russell’s intention to state the Tristram Shandy illustrates, via the use of the correspondence principle, that Tristram would be finished his book. Craig states, if Tristram Shandy were “immortal, then the entire book could be completed, since the method of correspondence each day would correspond to each year, and both are infinite.” (Craig, Kalam Cosmology. P.33). However, as stated above Russell actual only states that no day will remain unwritten, which in the sense of the potential infinite is true. Craig makes the point of stating this is a potential infinite so obviously Tristram will fall continually behind in his work. Craig believes there is a contradiction in his observation that Tristram will be continually behind and at the same time finished. However, the problem being alluded to here is the fact we are looking at a potential infinite. So at any given year of writing he will be behind and there will also be a finite number of years he has spent writing. So perhaps we should consider that a potential infinite does not really fit to the correspondence principle being used to show infinite sets are equivalent.
There are no problems with Russell’s Tristram Shandy illustration, if taken as a potentially infinite series. At no time is Tristram near completion even if the sets are increasing in terms in a one-to-one ratio. We can also say that no part of his autobiography will remain unwritten as time progresses in this potentially infinite series. There is a problem, perhaps, in the notion Tristram will ever reach the point to have written an actually infinite amount of years about an actually infinite amount of days since the very nature of the potential infinite is a series that never reaches its limit and with a first term no matter how long he has spent writing it is a finite span.
By applying the correspondence principle with the implication he has been writing an infinite amount of time about an infinite amount of years we then get let to the conclusion the infinite sets are equivalent and the whole is similar to the part. Which would be valid for actual infinites and still, given the sets are instantiated in time and one is far more extensive than the other, Tristram is still not finished his writing. Poor fellow still has an actually infinite amount of unrecorded days to get through. Talk about traversing the infinite. We must note that while the whole is similar to the part under these conditions, we still do not mean ‘every’ part. The only way we could say every part in this series instantiated in time if it took him a day to record a day. The set of recorded days and set of years spent writing are equivalent in number of terms, but clearly not in duration.
William Lane Craig will later flip this Tristram Shandy scenario around such that we are dealing with actual infinites which he then suggests does lead to paradoxes.
Craig, William Lane. The Kalam Cosmological Argurment. Macmillion Press Ltd: London and Basingstoke, 1979.
Russell, Bertrand. The Principle of Mathematics. W.W. Norton & Company, Inc., New York, NY, 1943.
Albert, Nikki. The Tristram Shandy Paradox. 2002