"Make it new" is, of course, the resonant phrase. Ezra Pound's famous words, so aggressively modern, became a key phrase for a generation of artists, art critics and curators. It stands for the work of two generations of artists, from Delaunay's Windows Open Simultaneously, Kandinsky's Untitled (First Abstract Watercolour) and Malevich's first Black Square, all painted within the decade leading up to 1915, to mid-20th-century figures like Miró, Mondrian, Rothko and Jackson Pollock.

It is no coincidence that these great works coincided with a new wave of art museums and collections: the Museum of Modern Art, America's premier museum devoted exclusively to modern art, and the first in Manhattan to exhibit European modernism, founded in 1929; the Musée National d'Art Moderne in Paris, founded in 1937, the first museum of its kind created in Europe; and the Museum of Non-Objective Painting, which opened its rented quarters in Manhattan on East 54th Street in 1939, the first home to the Guggenheim collection. It is also no coincidence that these three museums were in Paris and New York, the two great centres of abstract art for almost half a century.

What do ambitious new art museums need? They need a story about modern art. In particular, they need a story about progress. The old was past and the new (abstract art in all its forms) was the future. The old conventions were gone. The new, "the shock of the new" in Robert Hughes's phrase, was here to stay.

I spent the baking hot summer of 1976 in Paris, making perhaps a dozen visits to the Musée National d'Art Moderne and another dozen to the Jeu de Paume. The history of modern art could not have been more clearly laid out. There were the great Impressionists and Post-Impressionists at the Jeu de Paume. A brief walk away was the temple of high Modernism and abstraction, with all its many "isms": Futurism, Constructivism, Suprematism, Dadaism, Surrealism and on and on. Of course, there were nods to figurative artists. But, a handful apart, they were marginalised, pushed to the side-roads of modern art, while the great story was about the rise of abstract art, at the heart of Modernism.

In his seminal 1961 article "Modernist Painting", the influential American art critic Clement Greenberg wrote: "Modernism includes more than art and literature. By now it covers almost the whole of what is truly alive in our culture." Abstract art wasn't just one of many forms of artistic creativity. It became, in the mid-20th century, a byword for a form of sophistication and artistic fashion that figurative artists like Courbet, Chagall, Beckmann and Bomberg could never hope to be part of.

Abstract art wasn't just in museums. It was everywhere in the culture, on book covers, record covers and in films. Take some of the most iconic record covers of the Sixties and Seventies: The White Album by the Beatles (1968), Tubular Bells by Mike Oldfield and The Dark Side of the Moon by Pink Floyd (both 1973). Or the cover of the 1966 edition of Al Alvarez's poetry anthology, The New Poetry. It featured a Jackson Pollock painting because that's what "the new" meant. Or take the cover of the original Faber edition of The Bell Jar by Sylvia Plath in 1966. This cover became an icon of modern design. If something was serious and highbrow it had to have an abstract cover, whether it was the Fontana Modern Masters covers of the 1970s or Perry Anderson's books on Western Marxism with their covers from abstract paintings by Robert Natkin.