Manhattan's doodle dandy: Is there any depth to Saul Steinberg's cartoons?

Saul Steinberg was an original. He invented a new form of cartooning. You might call it conceptual cartooning, or self-referential cartooning, or cartooning-about-cartooning. He had his own style, true – a distinctive, wonky-elegant drawn line. But he also played with style, with numerous styles, and with the languages of cartooning and drawing generally. His images became a byword for visual sophistication, associated with one particular city, New York, and one magazine, The New Yorker, for which Steinberg worked from the 1940s to the 1990s. Steinberg had style.

Some people classed him as an artist. Unlike many 20th-century cartoonists, he wasn't afraid of modern art. He was fluent in it. He knew the New York scene, was friends with some of the Abstract Expressionists, made easy references to them. His own graphic games owed much to European artists of the previous generation – for example, to Duchamp, Magritte and Miro. His work appealed to the art-aware and appeared in galleries as well as in books and publications. You can see it in an art gallery now; about 100 of his original drawings are at the Dulwich Picture Gallery, where Saul Steinberg – Illuminations has opened.

Here's one, which shows how Steinberg sometimes needs the wall space of an exhibition. The Line is a strip drawing, 33 feet along, following the mutations of a continuous, straight, horizontal line. It becomes, in turn, a washing line, the top of a bridge, the wainscot of a room, the edge of a table, the water surface of a swimming pool seen in cross section, and the horizons of several kinds of landscape, before ending up as a plain line being drawn by a hand.

Before your very eyes, the image demonstrates the diverse ways in which a drawing can use line. It's characteristic of Steinberg's highly self-conscious procedures. He lays bare the working of pictures, showing how they are made of devices and conventions by mixing together codes that are normally kept separate. Print and handwriting, letters and numbers, doodling, scribbling, maps, decorative borders, speech balloons and thought bubbles, diagrams, punctuation marks and figurative drawing all find themselves in the same frame. Incompatibles interreact. The image self-deconstructs.

A drawn human figure unravels itself into a free, loopy doodle. Ornamental flourishes are planted out as a garden of flowers. Group Photo shows rows of heads and shoulders in suits and ties, but each head is an (identical?) fingerprint. From 1953, there's Techniques at a Party, showing a gathering of 18 guests, each realised in a different manner – very solid, very feint, very messy, pointillist, Picassoid, and so on – each portraying the guest's party personality. Bleecker Street is a freakville version of the same idea, set in Greenwich Village, illustrating the cacophony of Sixties bohemian identities. Style-as-character is one of his abiding tricks.

The animation of typefaces is another. In Erotica, the number five and a question mark lie curled up on a bed, having sex. Big words come out to play. The phrases I AM, I HAVE and I DO are set together in an existentialist drama. NOW confronts NEVER. YES hits BUT. Broadway is severe comment on New York's theatreland. It's a street whose buildings are five huge block letters. Each letter is inscribed with critics' superlatives, "fantastic", "magnificent", "dazzling", "sublime". But the five block letters spell out T R A S H.

Or there's map-play. Steinberg's most famous image is an example. View of the World from Ninth Avenue is a subjective map, showing the New Yorker's parochial awareness of the rest of the planet; 10th Avenue is full of detail, but beyond the Hudson river things start to foreshorten abruptly and mainland America is a narrow strip, the Pacific a narrow strip, Asia a peep. Elsewhere, tourists go on about their foreign holidays with maps as speech bubbles. The Flat Earth is a Mappa Mundi for the 20th century – the continents squared-off and flat-packed and streamlined.

It's an admirable body of work. If you don't know it, you ought to. It has proverbial status. Though I've never heard the name made an adjective, Steinbergian could be a shorthand term for pictures-about-pictures – just as Pirandellian is for plays-about-plays. And in its achievement, Steinberg's art deserves fame; it is intelligent, versatile, alert, apt, all that, unquestionably. Yes. But. (You saw that coming?) But I must admit that it leaves me fairly indifferent. I can praise, but I can't enthuse. Something is missing, and it's an important thing.

Steinberg's cartoons are witty, but never funny. There are sometimes figures with "funny" faces, but these figures are comic tokens. I can imagine a Steinberg image provoking a smile of satisfaction at its cleverness, and a smile of self-satisfaction at getting it. I can't see one raising a laugh.

Nor does his art ever become painful. It's often bringing up big, serious things. Steinberg is exercised by issues of self and motives, individual and society, time and space, life and death – but in an abstract form. In that party scene, the style-play puts the characters at a remove; it indicates their vanities and agonies, but keeps them from being felt. The big words remain big words.

Steinberg's pictures have a two-way get-out clause. They don't have to be troubling, because they're cartoons. But they don't have to be funny, because their concerns are deeply grave. Neither this nor that, they end up in a limbo of weightless ingenuity.

This is the trap of sophistication. Your first response to Steinberg's work is a sense of exhilarating freedom. Here's an art that, aware of all the rules, is bound by none. It exists in a state of complete detachment, irony, free play. Anything seems possible. Ah, if one could live like that, playing all the roles and all the games! It embodies the ideal of a free city life – and for all its critique of New York manners, Steinberg's art provided New Yorker readers with a very flattering mascot.

But this freedom becomes empty and programmatic. The artist is totally in control. He never meets reality, the resistance that can generate comedy or tragedy. He doesn't touch the ground. He's operating a self-contained world of signs. Which sounds like the scenario of one of Steinberg's own pictures. He often did cartoons about artists, their hubris and helplessness. His knowingness about the creative process was one of the things that got him called an artist himself – or even a creative who outflanked art by doing lightly and effortlessly what artists did ponderously and pretentiously. And there's sometimes truth in that.

But in general, the work of Steinberg should return us to the artists he borrowed from, with a strengthened admiration. It makes you see how much more is going on in Magritte, Miro, even Duchamp; how their games are imbued with a mystery, melancholy, violence, that you never find in Steinberg.

He does the same for the true cartoonists. Just a glimpse of Charles Addams's brooding malice, or Robert Crumb's bug-eyed grossness, or Roz Chast's beady-eyed fragility, is enough to show up Steinberg as a highfalutin straight man. Artist/cartoonist? Or falling between the two stools?

Saul Steinberg – Illuminations, Dulwich Picture Gallery, London SE21 (020-8693 5254; www.dulwichpicturegallery.org.uk), to 15 February