If it wasn't for the fact that she likes to make jokes - "I think of it as a bit of a defect" - Anne Carson wonders whether she might have become a serious philosopher. Instead, her books sit in the poetry section, where they generate mild outrage for failing to conform quite to genre. The subtitle of her latest volume, Decreation, is Poetry, Essays, Opera, and the one before that, The Beauty of the Husband, was described on the dust-jacket as "a fictional essay in 29 tangos". This seemed to cause pain in particular to a group of male poets from Canada, Carson's birthplace, and they convened on the internet to decry her "pretentiousness".

Carson is 56 and a heavyweight: the first woman to have won the TS Eliot Prize for Poetry, twice shortlisted for the National Book Critics Circle Award and made a MacArthur Fellow in 2000. She also sells very well for a poet, which is why, even though her work militates against almost every commercial principle in publishing - this is a woman who will happily devote 50 pages to discussing 14th-century French mysticism and round it off with a joke about Kant - her publisher, Knopf, leaves her pretty much alone. "Lucky," she says, and giggles.

She is a classicist by training, who after graduating from Toronto University taught Latin and Greek at Princeton and for the past three years has taught part-time at the University of Michigan in Ann Arbor, a bijoux town an hour's drive from Detroit. "I kind of rest in the margin of being foreign," she says, and it suits her nature. We sit on the balcony of the house that she shares with her boyfriend, a conceptual artist, Carson like a scholarly Joni Mitchell in cut-off denims and a billowy white shirt, and when she talks it's in a faint, hippy-ish voice that makes it hard to tell if she's joking.

Her books are like collages, a combination of memoir, poetry, dissertation and drama, held together each time by an overriding theme. The question of what formal category they fall into doesn't interest her. "You write what you want to write in the way that it has to be." The language is often acute. The opening page of The Beauty of the Husband is as arresting as any modern poetry I have read. "A wound gives off its own light / surgeons say / If all the lamps in this house / were turned out / you could dress this wound / by what shines from it." She can be surprisingly gossipy. In her latest book, in the poem "Gnosticism IV" (the title gives you an idea of how little interest Carson has in, say, making Oprah's Book Club) she asks readers to imagine the awfulness of an academic dinner with "Coetzee basking / icily across from you at the faculty table". What, as in JM Coetzee? She giggles. "Yes. That was unkind of me, but it's him. I met him once and I can't say he was unkind to me, he was very courtly, but his effect in general was odd. He was confrontationally aloof, if that's possible."

To give you an idea of how hard it is to describe what Carson does, you need to see a fullish running order of her latest book. Decreation starts with some sad, wry poems about her late mother - "to my mother / love / of my life, I describe what I had for brunch." Then two academic essays, one in praise of sleep, the other about the sublime as it appears in the work of Longinus, a first-century Greek essayist, and Michelangelo Antonioni, an Italian modernist film-maker. Then some poems about the sublime; then an "oratorio for five voices", called "Lots of Guns", which is very funny and was originally written as a tribute to Gertrude Stein (Carson reminds one a bit of Stein, the way she tries to make points about the nature of connectivity by sailing very close to randomness). Then an essay about eclipses; a screenplay recasting the medieval French lovers Heloise and Abelard as an American sitcom couple; and finally the main event, "Decreation", an "opera in three parts", which examines the work of three female mystics, Sappho, Marguerite Porete, who was burnt at the stake in 1310 for writing a heretical book, and the French philosopher Simone Weil. "We participate in the creation of the world by decreating ourselves," writes Carson.

Her critics accuse her of being wilfully obscure and she agrees with them up to a point, although she says that it's a question of personality rather than affectation. "I am kind of a curmudgeonly person, so I don't gravitate to groups or traditions, which is probably just pretentious of me." Her wacky juxtapositions (in her book Men in the Off Hours she imagines a dialogue between Virginia Woolf and Thucydides taking place on a TV panel show) are, depending on your view, either highly original and revealing or highly contrived.

Carson says she's not trying to show off; it's just the way her mind works. She is a messy writer: "[it's] a basket of stuff that eventually looks like it has some informing idea. Then I grope around in it to see what that is, try different orderings and different concepts and then fix on one." Classicists are probably more sensitive than most to the suspicion that no original thoughts are left to be had in the world, but in any case, Carson believes that thoughts themselves matter less than the routes one takes between them. "I don't know that we really think any thoughts; we think connections between thoughts. That's where the mind moves, that's what's new, and the thoughts themselves have probably been there in my head or lots of other people's heads for a long time. But the jumps between them are entirely at that moment." She says, "It's magical."

Originally she wanted to be an artist, and her first book of poetry, Short Talks (1992), began life as "a bunch of drawings which I put titles on, and then the titles got longer and longer" until the drawings disappeared. Her ideas still tend to come to her in shapes first and her poetry is very visual - she arranges it in crazy shapes on the page. I ask if they refer to anything or if it's possible for them not to refer to anything. "The shapes are [meaningless] I guess, although shapes aren't non-referential, but then it's not a shape of anything but itself." There is a long, confounded silence. "This is why we're not philosophers."

When Carson was a child she read a book called Lives of the Saints and loved it so much that she tried to eat the pages. It sounds like an apocryphal story, but yes, she says, "I did do that." Neither of her parents went to university. Her father had fought in the second world war and liked to read history books; her mother liked abbreviated versions of the classics sent by Reader's Digest. "We had zillions of those around the house. I used to read them, but they're not very satisfying ... We were always sitting around, the three of us, reading in a room in the evening. And my brother, who had a quite different personality, would come and stand in the doorway and say, 'I can't believe you people.'" She smiles. "He was into cars and girls and bars."

Carson took Latin in high school because it was the alternative to typing. Her Latin teacher was also conversant in ancient Greek, so Carson took Greek lessons in her lunch hour. "Greek is one of those things that, when you do it, you realise it's the best experience in the world, there's no reason ever to stop. It's just some amazing combination of the kind of puzzle-solving that goes into crosswords and amazing literature. You think, well, they're nerds, they were born that way. But they're not just nerds, they're all kinds of people who stumble into this happy field of endeavour and stay there." To her parents' alarm, she announced that she was going to pursue these two, entirely impractical dead languages at university. "My father kept telling me to get a marketable skill on the side. He suggested typing. He was worried for some time. And then I got a job at Princeton and he sort of calmed down."

If her study of Greek and Latin has affected her own writing style, Carson suspects it is to be found in the way she makes patterns between things. "There is something about the way that Greek poets, say Aeschylus, use metaphor that really attracts me. I don't think I can imitate it, but there's a density to it that I think I'm always trying to push towards in English. It's a kind of compacting of metaphor, without a concern for making sense ... it's just on the edge of sense and on the edge of the way language should operate."

The danger with this, and with Carson's writing, is that it drifts into whimsy or nonsense. "It does fall apart a lot. It gets just too weird for anyone to care about reading, or else it gets diluted into a sort of parody of itself. Intuition is the only way to keep on the line between them. And also focusing back on to the first time the idea came into your head has some kind of pristine conviction that it gradually loses." Carson returns to the actual piece of paper on which she wrote down the beginning of the idea, usually a coffee-stained back of an envelope. "Because there's something almost magically convincing about that piece of paper. The same words typed on a nice clean piece of paper wouldn't have whatever it is - fidelity, to your original thought."

You wonder what her parents make of it all. She remembers when her first book came out, Eros the Bittersweet, in 1986, which was based on her PhD thesis about Sappho. "My father was puzzled. My mother read it up to page 37, she turned down the corner and put it back on the shelf, intending to return to it, but never did. I used to look occasionally, in a casual way. After that, I don't know that they really read things diligently. I would send the books to my mother and she would put them on a shelf near the door and point proudly; but I don't think she really enjoyed them."

She was married once and wrote about the break-up in The Beauty of the Husband, including an account of how her husband spitefully stole her notebooks when he left. (He eventually sent them back.) The conventional, storytelling quality of Carson's poetry is so readable that I wonder if she has ever thought about writing a novel. "I tried that. Well, I had the aspiration, when I wrote Autobiography of Red, to write a regular novel, like one you would buy in an airport. I kind of started it as a dare and then it turned into that ... poetic thing." She laughs. (Autobiography of Red is in part an updated version of the myth of Geryon and Heracles.) You got bored? "Yeah, it just kept having too many words. When I get too many words, I don't feel that I'm saying anything. I'm just saying the words, not the thing. So I have to keep cutting it down, cutting it down, and it gets turned into verse. It was hopeless."

The academic and high-concept stuff in Carson's books seems at times to be a cloak for the humour, which she half- dismisses as frivolous and is a little embarrassed by - although these are the bits that you come away wanting more of. She continues to defy any one category. She has just translated a new version of four plays by Euripides; there is a plan to go to Germany to do some kind of performance poetry with her boyfriend, who looks nonplussed when she mentions it. Carson sighs. "Not knowing what one is doing is no prohibition on doing it. We all grope ahead."