There is something malodorous about this book - like the stinking petals of a rotting bloom. Gardner first published an annotated version of Carroll's Alice books in 1960, and since then he has - with a pedantic avidity that makes train-spotters appear lazily dilettante - continued to amass more and more material concerning them. This attractively produced and painstakingly edited update contains a long-buried, excised episode from Through the Looking Glass (entitled "The Wasp in the Wig" - although this also appeared in Gardner's intermediate More Annotated Alice  in the US), introductory essays for all three versions of his work, together with notes on Alice on the screen, on Carroll societies, on John Tenniel's illustrations (many of which appear here, faithfully reproduced), as well as the vital annotations themselves.
But, as I say, there is something not quite right about the work overall, something off-colour, off-key and distinctly iffy. Gardner himself is a valetudinarian (born in 1914), who spent a short lifetime (1952-82) writing a monthly, recreational mathematics column for Scientific American. He has written numerous other works on mathematical puzzles and conundrums and, on the flyleaf bio of this work, the philosopher Douglas Hofstadter hails him as "one of the great intellects produced in this country in this century" - whatever that means. Like my old uncle, Robert Ross, Gardner lives in Hendersonville, North Carolina; and, like Uncle Bob, I imagine him to be one of those Americans who present an image of curiously antiquated gentility - Gramercy Park clubbable, Brooks Brothers tailored, New York Times erudite - as if, like certain hill- billies in the Appalachians, they have retained elements of an England long vanished over here.
I mention all of this because I can only hope that it is Gardner's recondite notions of amour propre that lead him into an analytic cul-de-sac of such tedious brevity that it almost entirely compromised my interest in and enjoyment of this work. For Gardner staunchly maintains that Carroll's obsession with pre-pubescent girls, his photographing and sketching of them in the nude, his delight in kissing them, his revulsion from boys at a similar age, his apparent revulsion from adult sexuality of all forms - that none of this represents anything but "complete sexual innocence" on Carroll's part. Gardner is a man uncomfortable with the psychoanalytic; of such interpretations of the Alice books (which he disbars from his own annotations and sequestrates in the bibliography), he says: "We do not have to be told what it means to tumble down a rabbit hole or curl up inside a tiny house with one foot up the chimney. The rub is that any work of nonsense abounds with so many inviting symbols that you can start with any assumption you please about the author and easily build up an impressive case about it."
The problem is that Gardner's own analysis of Carroll proceeds via a false syllogism (a misapplication of logic that doubtless his subject - a pedantic and second-rate Oxford mathematics don - would have approved in his own defence, if not in his tutorials). For Gardner: a) all paedophiles manifestly wish to have sex with their objects of desire; b) there is no evidence that Carroll wished to have sex with Alice Liddell or his numerous other "child loves"; therefore, c) Carroll was not a paedophile.
Actually, this isn't even a false syllogism - it's just entirely false. Many paedophiles successfully hide the sexual nature of their interest in children behind other forms of engagement; but anyway, Carroll didn't - he photographed them naked. Gardner's uneasiness with the psychoanalytic, his inability even to engage with it, is part of his conflation of ignorance with innocence: we cannot see the erect penis in the "devout Anglican" Charles Dodgson's trousers as he photographs naked little girls, therefore it doesn't exist!
Gardner even goes so far as to suggest that, because there is no reference to Carroll in Lolita, Nabokov would have supported Gardner's own avowal of his "innocence". And yet, he belies this by quoting from an interview with Nabokov (himself a translator of the Alice books), where he spoke of Carroll's "pathetic affinity" with Humbert Humbert, adding that "some odd scruple prevented me from alluding in Lolita to his wretched perversion and to those ambiguous photographs he took in dim rooms". I'm with Vladimir on this: Carroll was indisputably a paedophile - just not an active one.
All of this is important. We don't want to make "any assumption" we please about Carroll's psyche; we wish to acknowledge one vital fact about his sexuality, and, without that, any annotation of the Alice books becomes an exercise in wilful distortion. Gardner notes that Carroll practised the White Queen's advice to Alice in Through the Looking Glass to "consider anything, only don't cry".
In Carroll's introduction to his little work Pillow Problems (pretty suggestive in itself), he speaks of working at mathematical problems during sleepless nights to prevent his dwelling on "unholy thoughts, which torture, with their hateful presence, the fancy that would fain be pure. Against all these some real mental work is a most helpful ally." And the mental work he most conspicuously indulged in to keep these "unholy thoughts" at bay was the children's books themselves, with their panoply of conundrums, devices, parodies and burlesques.
So, Gardner's annotations, with their obsessive dwelling upon the minutiae of factoids about the texts, rather than their symbolism, are themselves a gloss upon a displacement activity. What, in my view, makes the Alice books so enduringly central to the English literary canon is precisely their quality of heightened repression: the struggle by a tormented paedophile to keep the manifest object of his desire straitjacketed in a fallacious - yet socially condoned - dreamlike realm of sexual ignorance. It is this that made everything in the Alice books - from the distortions in scale, to the surreal elisions, to the banjaxed language - such a rich source of inference for those avatars of modernism, Joyce, Eliot and Nabokov.
I cannot fault Gardner on his detailing of the proceedings of the Carrollians: it is all here, from the whimsical observation that Tenniel's depiction of the Mad Hatter was a prolepsis of Bertrand Russell's face, to the Caterpillar's inquisition ("Who are you?") as a retailing of a mid-Victorian mass catchphrase (the "at the end of the day" of its time). And there's much more of little relevance, including complete versions of all the poems and songs Carroll parodied, details of who was who in all the film versions of Alice, and explications of terms still familiar to the modern English reader but obscure to Americans.
Gardner makes a strong case for the White Knight in Through the Looking Glass as a self-portrait of Carroll as a sad, wistful, chaste old man (he was 40 when he wrote it), saying goodbye to his love who is doomed to hormones. Personally, I wish Gardner had taken seriously the notion of Carroll as a dirty old man - then we'd have a truly adumbrated Alice, rather than an annotated one.
Will Self's most recent novel, How the Dead Live (Bloomsbury, £15.99), was shortlisted for the Whitbread Novel of the Year