If you're of a sensitive turn of mind, you might want to stay away from Tate Modern's new show, Duchamp, Man Ray, Picabia, on two counts. The first is that you will find your heart bleeding for the last member of the trio, and even, to a lesser extent, for the second. Then there is the infinitely depressing fact that the art of the leader of the pack – Marcel Duchamp, a man whose influence on Western culture arguably ranks with that of Freud and Darwin – looks as though it might have been made yesterday, which it could.
Unless you've made a hobby of the early 20th-century avant garde, you may wonder why the three men have been included in one show at all. The answer to that is that they planted the seeds of an anarchic movement, born in Zurich and known as Dada, in New York in the early years of the Great War. Dadaism done, the trio contributed variously to the various Surrealist currents in Paris and elsewhere, and remained friends and rivals for nearly 40 years until the Franco-Cuban Picabia's death in 1953. (Duchamp lived until 1968 and Man Ray, the baby of the group, until 1976.) The Tate's show begins with portraits of the three men of themselves and by each other. It ends, a dozen rooms later, with the aged Man Ray obsessively re-working Duchamp with an energy that seems part homage and part hatred.
This is all to say that Duchamp, Man Ray, Picabia sets out to suggest an equivalence between its protagonists, men it describes as "three of the great pioneering figures in early 20th-century art". From the outset, though, it is clear that this even-handedness is, to put it kindly, quixotic. To put it unkindly, it is nonsense.
The first rooms show the artists' juvenilia, their early careers having all been spent as painters. Even here, it's clear that we are dealing not with a three-legged race but with gold, silver and bronze. Duchamp's Nude Descending a Staircase, No 2 (1912) is an amazingly sophisticated work, synthesising Cubist and Futurist thinking in a way that is entirely confident, wholly its own. The Lovers (1914) by the American-born Man Ray, ex-Emmanuel Radnitzky, shows the artist's exposure to the European avant garde via the 1913 Armory Show. It also suggests a native way with form. Adam and Eve (1911) by Picabia – in this context, poor, sad Picabia – is flaccid and floundering; and, bar his Fig Leaf of 1922, that is how he stays for the long course of this show.
Over Francis Picabia's inclusion, in other words, hangs the question: why bother? His role in the theorising of Dada was certainly important, but his work itself is an embarrassing distraction from the main event, which is Duchamp. The same, though to a far lesser extent, is true of Man Ray. The cruel truth is that Duchamp's particular genius lay in uncertainty, and that uncertainty was (and is) the defining spirit of the age. Its greatest lyric poet was Sigmund Freud, who showed that humanity is defined by a sexuality revealed in accidents and slips of the tongue, and it is that revelation that is at the heart of Duchamp's work.
Duchamp dressed as a woman whose name – Rrose Sélavy (Eros, c'est la vie, roughly "sex is life") – was a punning celebration of the ambiguous. So, too, was his notorious drawing of a moustache on the Mona Lisa. His eye (and ear) for the accidental was miraculous, ranging from endless French games with homophones – puns on the words savant and savon, noting that the letters LHOOQ sound like "she has a hot arse" – to seeing a wine rack as a woman's body covered in penises. And Duchamp wasn't just a master of form. The means of his art-making – the spotting of visual kinships in those everyday objects he would dub "readymades" – was itself scrupulously random, the sporadic outcome of a sporadic time. Half a century before, Charles Baudelaire had lauded the artist of the future as a flâneur, a lounger in the streets. It is arguable that his prophecy found its fulfilment not in Manet, but in Duchamp.
I've saved that urinal for last for the good reason that it is arguably (and laughably) the most important artwork of the past 100 years. Fountain – inscribed with the pseudonym R Mutt – isn't entirely a thumbing of the nose – it also has a formal interest as a feminine object designed to receive the outpourings of men – but it is largely that. It was "made" barely a decade after Duchamp's late-Impressionist painting, House in the Wood, and yet, philosophically, the two works are separated by aeons.
Between Duchamp's Fountain and Tracey Emin's My Bed, though, there is not a day's difference. Both are about the same historical accident, namely the transporting of a non-artwork into an art context. Fountain – actually a Sixties copy, the original having been lost – will have been carried reverently by conservators to Tate Modern's Level Four in its own champfered crate, it will be insured for God knows how much, and we will gaze at it in awe. Layered by now with so many meanings and so much history, it is neither an artwork nor an artefact but a totem. It is a snug-bar truism to say that nothing new has been done in art since Duchamp, but, looking at Fountain, it is also true. The rest of this show is merely a footnote, and so, perhaps, is the rest of art history.
Tate Modern, London SE1 (020 7887 8888) to 26 May