In Notes from Overground, Tiresias (my former neighbour Roger Green) quotes with approval from a lecture by W. H. Auden, on “four questions which, could I examine a critic, I should ask him”. He said:
He said: ‘If a critic could truthfully answer “yes” to all four, then I should trust his judgement implicitly on all literary matters.’ The first (I will not bother here with the other three) was: ‘Do you like, and I really mean like, not approve of on principle, long lists of proper names such as the Old Testament genealogies or the catalogue of ships in the Iliad?’
His own comment on this is that “Short catalogues … have a magic, incantatory quality amounting to much more than the simple sum of their parts”.
Certainly Tiresias takes great pleasure in lists, not necessarily of proper names. By the time we reach Auden’s opinion, we have had lists of railway delays and commuters’ responses to them; trucks (one of his preoccupations, subdivided into trucks with fishy names, trucks with ungainly names, trucks with functional names, rolling-stock which defy categorisation, and legends on rolling-stock); public figures seen on the train; words and phrases defining commuters; reading-matter of his fellow commuters; unpleasant features of new inter-city trains; occupations of commuters, including “mathematician” (yours truly); birds seen from the train; and behaviour patterns of two types of commuter (which he characterises as sheep and goats).
One of the most famous examples is Jorge Luis Borges’ Celestial Emporium of Benevolent Knowledge from his essay “The Analytical Language of John Wilkins”. Wilkins had proposed a synthetic a priori language in which names encode the classifications — so that lists in such a language would be rather boring — and Borges invents his Chinese encyclopaedia to demonstrate the arbitrariness of such classifications. Borges gives as an example a classification of animals into fourteen types, delightful to read: the last two are “those that have just broken the flower vase” and “those that, at a distance, resemble flies”. Borges’ essay has been commented on by a number of philosophers and critics, a remarkable number of whom take it seriously. The Wikipedia entry gives a good account.
(Interestingly, the cryptographer William Friedman proposed that the mysterious Voynich manuscript is written in an “artificial or universal language of the a priori type”; he followed Galileo in making this prediction in an anagram which was only revealed after his death. My source for this is The Voynich Manuscript, by Gerry Kennedy and Rob Churchill.)
A dictionary is, of course, a list on a grand scale; I am not alone in taking great pleasure in browsing in a dictionary, especially round about a word which I have just looked up. Lexicography is replete with lists. A very good source is Fowler’s Modern English Usage (I have access to the third edition, edited by R. W. Burchfield, fellow antipodean Rhodes Scholar, and former editor of the Oxford dictionaries). The book is alphabetical, and lists under the letter A include abbreviations; words ending in -able or -ible; adjectives used as nouns, with their first recorded appearance with each function in English; æ and œ ligatures and their treatment in British and American English; and words in -ative and/or -ive. But many of the lists are implicit, such as verse forms (sestina, triolet, etc.), which have individual entries but clearly form a category, or (probably the largest category in the book) distinctions between similar words such as “institute” and “institution”, or — one famously quoted by Churchill — “intense” and “intensive”).
Of course, I have mostly given examples above of lists of lists. Could there be a list of all lists? That way, presumably, lies Russell’s paradox.
So finally, after all that confectionery, we reach something resembling mathematics. What is a list?
In introductory probability, one meets a two-way classification of sampling: order significant or not significant; repetition allowed or not allowed. A set can be regarded as an unordered sample with repetition not allowed, and a multiset is an unordered sample with repetition allowed (so that we have to give the multiplicities of its elements, rather than simply their presence or absence in the collection). It is quite common to think of a list as an ordered sample with repetition allowed, and I think this is the commonest meaning in computing.
I am not quite happy with that. The examples I have given above, such as the list of lists occurring before page 44 of Notes from Overground, is ordered, but the order is quite arbitrary (it is simply the order they happen to occur in the book); I could have ordered them thematically or randomly. A dictionary is ordered lexicographically, but I feel that this is more to do with its function than its essential nature.
As I discussed here, “paper, scissors, stone” has an ordering which is not fixed but not arbitrary either; we could say “scissors, stone, paper”, but not “paper, stone, scissors”. Such a list is a set (or multiset) with a collection of permissible orderings. I argued in the cited post that it is sometimes useful to require the collection of orderings (regarded as permutations) to be a subgroup of the symmetric group of all permutations (as it is in the example, being the cyclic group). But I don’t think this is wide enough, even without the restriction to subgroups.
I don’t have an alternative to propose.