In March 1946 Pablo Picasso paid one of his fortnightly visits to see Henri Matisse in Vence, a few miles inland from Nice. Five years after the medical crisis that had nearly killed him, Matisse, at 76, was still an invalid: he had endured radical colon surgery and much of his work was now done either from a wheelchair or in bed. But, creatively, he was phenomenally fertile in the midst of what he called, with undisguised gratitude and wonder, “his second life”.

The evidence of this resurrection was almost entirely on paper: illustrative drawings done for books of poetry, Renaissance and modern and, most arrestingly, glued or pinned to the walls, the radically new form of modernism that Matisse had just invented – the cut-outs, an extended series of which would be published as Jazz by his friend, the critic and publisher Tériade, in 1947.

As usual Matisse was pleased to see Picasso but, also as usual, he was under no illusions about the purity of his friend and rival’s motives. In a letter to his son Pierre, dated March 19, he wrote: “Three or four days ago Picasso came to see me with a very pretty young woman [his lover and muse, Françoise Gilot]. He could not have been more friendly and said he would come back and have a lot of things to tell me . . . He saw what he wanted to see – my works in cut paper, my new paintings, the painted door, etc. That’s all he wanted. He will put it all to good use in time. Picasso is not straightforward. Everyone has known that the last 40 years.”

Matisse had long been conscious of Picasso’s magpie-like habit of thievery, which he often interpreted as a backhanded compliment or else as a spur to the mutual competition they had been conducting for almost half a century. At other times he didn’t like it at all. In 1926 Matisse wrote to his daughter Marguerite: “I haven’t seen Picasso for years . . . I don’t care to see him again . . . he is a bandit waiting in ambush.” Picasso would have been the first to own up to that larceny. As a boy, the only verb he had underlined in his school Latin book was latrocinor: “I plunder; I serve as pirate, robber . . . ”

Picasso and Matisse’s relationship was by turns comradely, suspicious and contentious; a kind of classic sparring match between Archilochus’s hedgehog and fox; the master who knew one great thing (let us call it, as Matisse did, “decoration”) and the master who knew many things; the essentialist and the encyclopedist. What did Picasso, the encyclopedist, think of Matisse’s most essentialist medium: the cut-outs? Though I don’t believe he’s on record at the time in offering explicit judgments, we do know that whenever Picasso used the word “decorative” it was pejorative, often to sneer at the Renaissance and Baroque masters he most disliked – Raphael (whose work he claimed no one would buy if it was offered afresh in the 20th century), Titian, Tintoretto, Rubens and, bizarrely, Caravaggio, all of whom he also wrote off as garishly “cinematic”.

Matisse, on the other hand, had spent much of his life arguing for the seriousness of the decorative and for the moral integrity of pleasure. “The decorative for a work of art is an exceptionally precious thing. It is an essential quality.”

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Matisse believed in the organic connection between decorative form and the irrepressibility of nature. He had committed himself to finding a visual language that would distil and translate the experience of pleasure into images with no loss of sensory intensity; that would, in fact, act as a kind of memory trigger of earthly delight.

After his last bout of maladies and surgeries, he believed he had miraculously found just such a visual economy of pleasure. But he was also defensive about how the cut-outs would be received. There were, indeed, old admirers and collaborators, such as the art historian Christian Zervos, who dismissed the cut-outs as embarrassing schoolroom hobbyism that no one would have taken seriously had they not come from the palsied hand of the master.

Zervos was a close friend of Picasso’s, so it is possible that Picasso subscribed to that damning judgment. There was, however, some evidence, both before and after Matisse’s death in 1954, of Picasso being haunted by the apparently effortless simplicity and vitality of the cut-outs. The most obvious instance was his “La Paix”, (1952) a huge mural, which was a nod to his friend, recalling as it did Matisse’s “Le Bonheur de vivre” of almost half a century before. So much of the career since was epitomised in the flat stylised fruit tree; the fertile sun; even the goldfish bowl; but which in its snipped-up and stapled-together look evidently aimed for a cut-out aesthetic. Although “La Paix” makes a valiant attempt at boyish romping, Picasso had to labour for innocence, whereas for Matisse it was like breathing.

After Matisse’s death, Picasso turned towards what came most naturally to his learned, pugnacious personality: playtime in the pantheon; riffing on the masters. A year earlier, Picasso, at 71, had been in crisis. Françoise Gilot had had the gall to walk out on him. There was the sense – inevitable for an artist for whom the erotic and the creative act amounted to much the same thing – that failure with Gilot had also brought him to a painterly dead end. Around Christmas 1953, she came to take their two children away while avoiding seeing Picasso himself, an event that triggered a moment of hitherto uncharacteristic self-pity, the swagger emptied of erotic and thus generative power.

The way he recovered was to find a project in which he could reassert the aggressive masculinity that he still boyishly felt was directly connected to his creativity; a project big with historical substance but one in which he could engage with the past masters in a spirit of mutual challenge. Picasso turned to an alternative remembrance: the history of painterliness. The passing of Matisse, the supreme modernist of the painterly mark, was still fresh, and he was struggling for some way to pay homage or assimilate Matisse’s instinct for joyous vitality without producing the kind of laboured pastiche of “La Paix”. What followed over the next 20 years – and it’s worth emphasising just how lengthily sustained the obsession was – was a protracted tussle with the past masters: Velázquez, Manet, Cranach, Poussin, David, El Greco, Degas and finally, and most poignantly, Rembrandt.

Picasso’s extended dalliance with the masters was hardly reverent. There was something of a fists-up attitude; Picasso taking them on to take them down. In a droll gesture of backhanded hat-doffing Picasso referred to Velázquez as “that bastard”. Picasso thought of his creative curiosity as having begun with a trip to Madrid in 1897 when he was 16, and he had been haunted ever since by the sly intelligence, the protean slipperiness of the painter. Like Picasso himself, there was nothing the 17th-century master couldn’t do, from the hyper-real and the classically linear to the freest and most suggestive marks; no genre boundary that he had been afraid to cross, no convention he wouldn’t disturb.

In response to questions about his fixation on the art of the past, Picasso replied that as far as he was concerned there wasn’t any; that any art that couldn’t actually live in the present ought not to be considered at all. The truth was, Picasso’s late historicism was really about him and his own increasingly panicky sense of obsolescence; a fear that especially on the other side of the ocean his kind of work was being trapped in a kind of pincer movement. On the one flank by Marcel Duchamp’s dismantling of the boundaries between the commonplace and the aesthetically loaded object, and on the other by the high-minded minimalism of abstraction; its insistence that true painting should be nothing more than the sum product of its own materials.

Against both of those temerities Picasso responded – admirably – by saying, in effect, don’t kid yourselves. The frame counts. Arbitrary art is a contradiction in terms, a declaration of suicide. As for painting: it can never be truly closed off from memory, autobiography, the vexed history of its own problems. It’s a response, then, indivisibly, to Picasso’s sense of his own last act – but also that of the modernism he had largely, if certainly not single-handedly, invented, one that had annihilated the simplicities of representation without ever doing away with the world.

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Matisse had concentrated on inventing an art that was all about the suspension of time; one that could capture momentary sensation and, through suggestive patterning (the suggestiveness requiring the active collaboration of the invited eye), indefinitely prolong that sensation, rather as if he had depressed the sustaining pedal of a piano.

This could not have been a more different conception of permanence from Picasso’s rummaging around the canon. Instead of the pantheon there was pantheism: Matisse’s belief in the perennial organic vitality of nature; in, you might say, its resurrectionary power. It was the nearest thing to formal religious conviction he had.

For Matisse, solid form was just the casing within which lay the true pulse: of colour and flyaway line. It was those qualities that gave decoration the force of therapy. Clinically. While for Picasso robust virility had, over his long life, been the norm, Matisse had signposted his own journey through art with episodes of traumatic sickness (either his own or the grave illnesses that laid low his wife Amélie). It had been while as a young law clerk Matisse had been convalescing after one of those illnesses that he had first been introduced to “chromos” – mass-produced colour reproductions, often of famous paintings – after which he went out and bought his first box of colours to copy them. Not long afterwards he met the draughtsmen who worked for a local textile factory at St-Quentin, who opened his mind to the possibility that stylised, flattened passages of colour could operate on the imagination, indeed on the whole metabolism, therapeutically, helping restore to it the notional golden mean of serene equilibrium.

Before the dramatic breakdown in his health that led to major surgery in 1941, Matisse had, however, been suffering (not unlike Picasso) from a sense of imaginative exhaustion, and Hitler and the blitzkrieg did nothing to make that headache go away. The urge to flatten had succeeded all too well. Looking at the work of the 1930s one picks up the ennui – a sense of tired replay – that would affect Picasso a decade later. The medical ordeal that Matisse barely survived suspended these conceptual conundrums. When Matisse got back to work it was with so profound a sense of grateful urgency that his agonising about drawing and painting now seemed, to him, in retrospect, like petulant self-indulgence. “My terrible operation has completely rejuvenated and made a philosopher of me. I had so completely prepared for my exit from life that it seems to me that I am in a second life.”

Even if Matisse, while he was doing the cut-outs for Jazz, didn’t quite know where he was going or what he was doing with them, he certainly did have something to search for. That something was the sign language by which the memory of sensations could be expressed without recourse to any kind of mimetic description – except of the loosest, most analogous kind. He became more and more enamoured of the cut-out, which differed from, say, a symbolic or emblematic visual vocabulary in somehow distilling the essence of something; the experienced sensation of its presence – a nude, a jellyfish, a tobogganist – down to its essentials.

Matisse claimed that he might study whatever it was he had in mind for a cut-out for hours, days, however long it took, before he was ready with the scissors; so that the working procedure became a happy succession of meditative calculation and dynamic physical impulse. Forms of locomotion other than the pedestrian kind of which he was now physically incapable recur in the cut-outs: swimming, of course, but also flight, both of which engendered visual experiences that were, in the best sense, untethered, weightless, and in which light, space, shape, volume and mass all had to be adjusted, or rather were never finally fixed and determined. It was not just the forms that he represented accurately as in gentle, organic, kinetic motion; in that shadowless light, it was the nature of vision itself.

When he finally got going with the scissors, blades accelerating or decelerating with the variable resistance of the stock, he did, indeed, take off: “I would say it’s the graphic, linear equivalent of the sensation of flight,” he said.

What the old man was also bringing into being were memories, but memories of an entirely different order from Picasso’s archive of the dilemmas of representation – along with his place in it. Sometimes the memories were remote. Jazz, Matisse made clear, recovered the experiences of childhood, the circus and travel: the forms discovered in the process of recollection somehow both vivid and ambiguous – the cowboy that is close to a happenstance Rorschach blot; the wolf that has the playfulness of fairy tales, which are, after all, the most terrifying stories of all.

The memories could also be mythic or remote. When Matisse came back from Tahiti (he travelled there in 1930, aged 60) he was surprised that his experience of the South Pacific had given him absolutely no visual inspiration whatsoever. Filtered through memory however, and translated into submarine realms of gold, the jellyfish and the fronded sea anemones play across the oceanic visual field.

Picasso with a portrait of Matisse at his pottery workshop in Vallauris in 1948

This freedom, not just from easel painting, but from the containing edge, the frame, was what Matisse sought from the play of forms he had somehow brought into independent organic life, shapes that embodied the forms of nature, without either laboriously imitating or departing entirely from their visual and tactile presence. So the termini of designed space were joyously over-run; the distinction between figure and ground made ambiguous (especially when Matisse incorporated the discarded shapes from a cut-out into the same composition). He described the correspondence between the play of those shapes and whatever had provoked their visual genesis as a “rapport”; an affinity that he then went on to say was, in fact, love, and “without that love there can no longer be any dependable criteria of observation and therefore no longer any art.”

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Speaking of love and art in the same sentence, we can safely assume, would have made absolutely no sense whatsoever to Picasso. Unless, that is, it acted as a euphemism for desire, which was another matter altogether. So when Eros, in 1965, had a serious brush with Thanatos in the shape of major surgery – either on Picasso’s prostate or his colon or both – the effect on his art was predictably traumatic. Recovering slowly, Picasso, for the last years of his life (he died in spring 1973) was in combat mode. Emotional nihilism, as much a shout of rage at his impotence as he can manage, doesn’t provoke Picasso to any serious act of reinvention, one comparable with Matisse’s scissorwork. Instead, he recycles his most aggressive distortions of the 1930s, only with a vengeance; even if, beneath all the raw fury, the layering of the paint is as seductive as ever.

It is on the etching plate, rather, that Picasso gets hard bitten, the acid doing the biting for him. And it is here that Picasso finally summons his most unlikely companion in voyeurism: Rembrandt van Rijn. As first noted by the Picasso expert Janie Cohen, the aged artist joined a long list of painterly predecessors obsessed with Rembrandt, speaking of him all the time; often signing books of his works as “Rembrandt” and projecting a slide of “The Night Watch” on his studio wall at Mougins, in the south of France, for days on end.

In his etchings, Picasso projected the virility of his past. The images alternate unnervingly between braggadocio and self-mockery. In one of the most elaborate of the etchings, which Picasso called “The Theatre of Picasso”, Rembrandt’s stagy “Ecce Homo”, showing Christ displayed to the people by Pilate, was blasphemously made over as the pantomime of the artist’s own erotic history; Picasso appearing as a gnomic version of the Saviour seated front and centre stage; with the crowd calling for his immolation now made up of a jostling audience of his wives and lovers.

At the end, then, Picasso was incapable of ridding himself of his multiple histories – erotic, aesthetic, thematic – which all collided, collapsed and then subsided in on themselves. What was evidently quite inconceivable was that he should share Matisse’s wish that before he died he should recover the innocent gaze, to “see the world again” as the painter said “through the eyes of a child”.

Except once, right at the very end. It was in the choice he made of a painting that would serve as the poster for an exhibition of recent work to be held in May 1973 in Avignon. Picasso didn’t live to see the show that was, for the most part, greeted with embarrassed silence or the rolling of reviewers’ eyes. But there was that poster: not, to be sure, the vision of a child but an image of one, or at least of a youth, wide-eyed; raptly holding on to brushes and palette. It was Picasso’s version of a little panel by the young Rembrandt (now in Boston) in which a doll-like figure is in the grip of a creative trance; the light of his idea fiery on the edge of the hidden panel. It was also Picasso’s last version of himself; the memory of a moment when everything was input not output; and the strokes of his brush were avid with possibility.

Simon Schama is an FT contributing editor

‘Henri Matisse: The Cut-Outs’ is at Tate Modern, London, from April 17 to September 7, tate.org.uk, and at the Museum of Modern Art, New York, from October 14 to February 9 2015, moma.org. The global sponsor is Bank of America Merrill Lynch