The two most shockingly physical, viscerally wounding works of art on view in current exhibitions in London are Damien Hirst’s bisected shark, “Death Explained” (2007), at White Cube, and Rembrandt’s “The Anatomy Lesson of Dr Nicolaes Tulp” (1632) at the National Gallery’s superb new Dutch Portraits show. Both involve the cutting up of dead meat. Both are meditations on life and death. Both play on the gruesome contrast between shrunken, inanimate, cold flesh and warm, colourful, breathing humanity. And both implicate the viewer. You are the living being who walks between the two glass tanks containing Hirst’s split shark carcase. You, watching Dr Tulp wield his scalpel, join the audience for the dissection of a hanged criminal.

“Death Explained” and “The Anatomy Lesson” are works by brilliant young artists who invented a new language for art’s oldest concerns and, in doing so, changed the course of its future. To see Hirst’s Beyond Belief show within half an hour of Dutch Portraits is to perceive how great conceptual art, like any significantly original work, at once builds on and smashes tradition. To evoke a full response, contemporary art and Old Masters need each other. London this summer eloquently juxtaposes the two.

Rembrandt was 25 when Dr Tulp gave him the commission that launched his career. Tulp was the star of Amsterdam’s surgeon’s guild and Rembrandt’s group portrait, depicting eight dignitaries, gave the doctor, charismatic in his floppy hat and flamboyant gestures, prominence and undreamt-of fame. Tulp also got more than he bargained for in terms of sensationalism. Draw close, and you notice that while one of the cadaver’s arms is already a mass of blood and muscle, the other, still intact, is a subtly different colour from the rest of the harshly lit body. X-rays recently revealed that it did not start out as an arm at all but a stump, the victim’s hand having been chopped off as a punishment for theft. Rembrandt altered the detail when the stump proved too harsh an extreme of Dutch realism even for patrons schooled to see art as man’s mirror on God’s nature.

Tulp probably had in mind something like Nicolaes Pickenoy’s “The Osteology Lesson of Dr Sebastiaen Egbertsz”, another group portrait focused on doctor and dead body – this time the skeleton of an executed English pirate. Hanging next to “The Anatomy Lesson”, the limitations of this accomplished, vivacious work, completed a decade earlier, throw into relief Rembrandt’s startling innovations.

Decisive among these were his adaptation from the Italian baroque of the chiaroscuro drama of brightness and shadow, and an overwhelming concern for the individual figure as it emerges – humanly complex, a feeling, thinking being – out of darkness into a spotlight both physical and spiritual. You can pinpoint the birth of modern man’s sense of himself – rather than God – as sovereign being anywhere from the Renaissance onward but no artist rivals Rembrandt for psychological truth joined to spiritual majesty.

The argument of this enticing show is that early in the 17th century, the northern Netherlands, newly liberated from Habsburg domination, initiated a form of intimate portraiture that, celebrating the rise of the bourgeoisie at the dawn of the capitalist age, would have been unimaginable to earlier societies and gives the Dutch a vital role in the progress to modernity. Compare Frans Hals’ sexy, casual 1622 marriage portrait “Isaac Massa and Beatrix van der Laen”, the couples’ smiles the “lineaments of gratified desire” as they recline together in a pastoral setting, with the hieratic, impersonal formalism of Jan Claesz’s 1602 family group “Albert Sonck and his son Frans, Lysbeth Walichsdr and her daughter Elisabeth”, and you see a quiet revolution in domestic consciousness.

Hals’ sparkling highlights and audaciously loose brushwork were the perfect style for this informal sensibility, fixing the spontaneous, snapshot moment: the happy child and nurse in “Portrait of Catharina Hooft”, nonchalantly posed “Willem van Heythuysen”, laughing, lace-collared “Pieter van den Broecke”, hand lightly resting on his cane – hands, wrote Van Gogh, “that lived, but were not finished in the sense that they demand nowadays. And heads too – eyes, nose, mouth done with a single stroke of the brush without any retouching whatever. To paint in one rush, as much as possible in one rush.” Every work by Hals here is impetuous, risky, an art of brinkmanship. “They say he was a drunkard, a coarse fellow,” said the dying Whistler when he visited Haarlem to pay homage. “Don’t you believe it. Just imagine a drunkard doing those beautiful things!”

Hals is the pioneer. Around him surges an enormous groundswell of talent and diversity – Salomon de Bray’s depiction of twin babies; Govert Flinck’s lavish likeness of Tulp’s daughter Margaretha; Jan Steen’s delightful genre painting of adopted child Bernadina van Raesfelt in a bustling poultry yard whose birds symbolise adoption and guardianship. And then comes Rembrandt. Four of the portraits here were completed before he was 30 and already range from the youthful, energetic confidence of golden soldier “Joris de Caulerij”, stepping out from the shadows in a blaze of light dizzyingly reflected on the hard metal of his gorget and the soft leather of his tunic, to the gnarled but lively features of brewer’s widow “Aechje Claesdr”.

No artist ever portrayed old age with more sympathetic rigour or exactitude. “Jacob Trip” is as luminous and other-worldly as a biblical patriarch. “Margaretha de Geer” is fragile, sad, dignified, impressive. “Portrait of an Elderly Man”, the corpulent subject slumped in a chair barely able to contain him, is a late life force, its expressiveness animated by the contrasting brushwork of Rembrandt’s final period: face almost moulded in three dimensions with palette knife and the back of the brush; hands, cuff and collars set down with a single robust stroke; thinly painted dark clothing scratched through with wet paint to reveal a glimmering ground. The balance is masterly yet the effect is not calculated but sketchy, free, unresolved as looking death in the face: a shattering high point of a great show.

‘Dutch Portraits, The Age of Rembrandt and Frans Hals’, National Gallery, London WC2, to September 16, sponsored by Royal Dutch Shell, tel: +44 (0)870 906 3891

A dazzling Grim Reaper

In 1991, Damien Hirst was 25 – the same age as the Rembrandt of “Dr Tulp” – when Charles Saatchi paid £50,000 for a shark in formaldehyde in a vitrine, called “The Physical Impossibility of Death in the Mind of Someone Living”. It was the defining moment when conceptual art acquired a new late-20th-century gravitas and celebrity, on which Hirst’s “Natural History” works have built ever since.

That series is wonderfully enriched by pieces from 2004 to 2007 in White Cube’s solo show Beyond Belief. The divided shark of “Death Explained” is the most direct echo and also the most disorienting because its parts appear to move queasily – an optical illusion caused by the water in the tank – as you walk between them. But everything here destabilises, for everything turns on wild reversals of spirit and matter, beauty and ugliness. A bullock pierced by arrows is a savage St Sebastian. “What Goes Around Comes Around”, a mirrored vitrine of units each containing a fish in formaldehyde, looks at first like an elegant piece of geometry; walk around the other side, though, and all the fish are skeletons. In a mock “Adoration”, sheep kneel in a neo-natal unit containing heart monitor, incubator and an infant cast in silver: biblical symbolism reduced to absurd materiality.

Like many artists whose images of ugliness and brutality initially shocked – Picasso, Bacon – Hirst’s aestheticisation of death is becoming more beautiful with time, partly as we come to accept the authority of his vocabulary. I don’t mean the dire, fatigued spot and butterfly paintings, which occupied him – or his assistants – recently; these will now appear a (lucrative) lacuna between the important 1990s work and this inspired show, where painting carries a new weight.

It’s cleverly done: Hirst sets up a duality between an abysmal group of graphically detailed, photo-based paintings, executed by assistants, on the subject of the birth of his son Cyrus by caesarean section, and painted in muted greens and greys, and a gorgeous series of huge, swirling abstractions in glossy pinks, purples, turquoises, speared with shards of glass and razor blades, which turn out to be based on textbook illustrations of cancer. Bacon’s obsession with medical treatises on mouth diseases comes to mind but so do Klimt’s tumbling, glittery, decadent images of birth and death, as well as expressive abstraction from Kandinsky to Pollock.

Hirst’s is a bleak, contemporary vision: high-tech birth sober with evocations of mortality; cancer – the most feared modern Grim Reaper – seductive as high art. These paintings are essential ballast to an installation culminating in the work that makes Beyond Belief the landmark exhibition of the decade, and Hirst still the most enthralling artist of his generation. Much has been written about his attempt to appal in an age beyond sensationalism, but “For the Love of God”, the platinum skull encrusted with 8,601 pavé-set diamonds, does something more difficult: it returns the viewer to an ancient awe for an art object almost primitive in its intensity.

The work itself references artefacts as varied as Aztec skulls – Hirst spends three months a year in Mexico – and Dutch vanitas. In the two minutes upstairs at White Cube, in that 21st-century chiaroscuro moment, however, with the spot-lit, scintillating icon of death perched on black velvet in a darkened room, bags and coats – quotidian existence – left behind in the corridor and voices dropping to a whisper, it is impossible not to recall the menacing blackness of Russian churches, where icons mediate between death-dealing God and mortal man.

Of course, the irony is that the gasps, whispers and hype are about money and bling rather than the skull’s vulnerable, deathly beauty. “For the Love of God” bites the spiralling art market that feeds Hirst; an emblem for our era of excess, it is a memento mori, in the month Blair leaves office, for a decade of showy insincerity and greed. But it is crazily democratic too: the most expensive piece of jewellery commissioned since the Crown Jewels, and made not for a czar but for any punter able to pay the £50m price tag. That roots the skull in our epoch. At no time before would such a thing have been economically or conceptually possible. Yet it will reverberate as art because it also asks, in a mad new way, the same questions as Rembrandt and Hals did about what it means to be human.

‘Beyond Belief’ is at White Cube, London W1 and N1, to July 7, tel: +44 (0)20 7930 5373