Entering through the Palladian portals of Burlington House, home of the Royal Academy of Arts, London, past the statue of Sir Joshua Reynolds, hangs a banner showing Édouard Manet's portrait, Berthe Morisot with a Bouquet of Violets . Paradoxically, the most radical artist of the 19th century craved official recognition – nothing would have given Manet greater pleasure than this exhibition, in the heart of the academy.

Gustave Courbet was the iconoclast of representational art, smashing through the barrier of subject matter. Proper subject matter – historical, religious, mythological – was set by the academies, as if in direct line of descent from the Renaissance and the Reformation, if not Greece and Rome. Suddenly, in 1850, Courbet's A Burial At Ornans , as Sarah Faunce sees it, "thrust itself into the grand tradition of history painting, like an upstart in dirty boots crashing a genteel party, and in terms of that tradition it was of course found wanting".
But Manet's modernist aesthetic was so far ahead of Courbet's that his work gave rise to vituperation. His Olympia (1863) was described as "the Hottentot venus with the black cat exposed completely naked on her bed like a corpse on the counters of the morgue". Besides being a complex visual treatise on French sexuality, Olympia was a portrait of Victorine Meurent, herself a singer, painter and model, who also appears in the most radical of Manet's paintings, Le Dejeuner sur l'Herbe .

Manet used the specificity of the portrait to comment on society, and this exhibition focuses on his portraits as never before. Unfortunately, Olympia and A Bar at the Folies-Bergère – paintings that are also portraits – are missing from the exhibition. If the portrait form is the carrier of meaning, their absence is a significant lacuna.

That said, viewing Manet's portraits to understand his Paris and its people is valid. In his review of the salon of 1846, the poet and critic Charles Baudelaire declared: "The spectacle of elegant life and of the thousands of floating existences which drift about in the underworld of a big city – criminals and kept women . . . all prove that we have only to open our eyes to know our heroism." Baudelaire and Manet were seduced by the new types of city dwellers, attuned to what they understood as the ironic heroism of modernity.

In his search for the "painter of modern life", Baudelaire mused on the tensions between tradition and modernity. From 1853, the demolition of huge swathes of Paris, and the construction of a new city under Georges Haussmann, was in train. With the city's history of bloody revolutions – 1789, 1848, 1871 – the authorities were determined to redesign with surveillance in mind. Large boulevards would put paid to anti-authoritarian behaviour, while the new 85 miles of wide, straight streets created a culture of consumerism and capitalism.

Between 1851 and 1872, the population grew from one million to 1.8 million. But in the process of modernisation, more than a third of the population was displaced and alienated. The building of hotels, railways, theatres, cafes, shopping emporia, parks, monuments, sanitation systems, churches, barracks and schools inevitably meant the dissolution of communities. In 1863, the year Manet shot to fame, 118,000 people were designated indigent, 10,000 were arrested as vagabonds, and more than a million were living in poverty and hunger. And in the city, the previously invisible, described by Balzac as “its sinister populations”, came out of the cracks in the old narrow winding streets, and joined the bustle and the buzz of the leisure metropolis and the technological new world that was the new Paris.

Manet, the dandy, lived the life of the flâneur or "saunterer", searching for social types. His gallery of modern life – actors, writers, musicians, gypsies, prostitutes, poets, absinthe drinkers, beggars, ragpickers, laundry workers, painters, politicians, family and friends – became the subject matter of his art.

In the tradition of 17th-century Dutch group portraiture, Music in the Tuileries Gardens (1862) includes family members, friends and major cultural figures (and a rare self-portrait): Zacharie Astruc, Jacques Offenbach, Henri Fantin Latour, Théophile Gautier and Charles Baudelaire. But here the subject is ultra modern, and the style experimental, as Manet paints in blocks of colour and works with broken brushwork.

Aesthetically, apart from radicalised subject matter, what shocked Manet's audiences was his technical audacity. The relative shallowness of the picture space emphasised a feeling both of proximity and self-awareness on the spectator's part, as the viewer is drawn close to the picture plane. Critics read Manet's accentuation of the flatness of the picture surface as a failure of technique; they could not understand an artist who appeared to cultivate ineptitude.
In 1863, his Déjeuner sur l'Herbe , shown at the Salon des Réfusés to outraged audiences, was a watershed. Inappropriate juxtapositions were noted, and he was criticised for poor composition. But these features were exactly what Manet wanted: to bring the unexpected and the contradictory together, to show the complexity of modern life. Although the painting alluded to the tradition of the fête champêtre garden party, in which the dressed and undressed were gracefully combined, Manet's Déjeuner discomfited audiences with its contrasts: its nudity and dress, its idealised imagery of the past from the uncomfortable reality of the present, its absence of a moral message and its critique of French social views on women and sexuality.

Manet was so far ahead of his contemporaries in drawing attention to art as art, and not, as hitherto, a transparent window on the world. His bravura handling of pigment led the way to modern, even non-representational art. Notwithstanding his painterly references to Rubens, Velázquez and Titian, he marks an abrupt rupture with the past, technically, conceptually and aesthetically.

The Portrait of Zola (1868) shows Zola at his desk, with his accoutrements as a writer and a critic. Manet and Zola were very close and Zola's support, as a friend and a critic, was crucial to Manet in the face of censure and rejection. Intriguingly, Zola wrote a piece about the experience of being a model that was reproduced as a pamphlet in 1867, visible on the desk. Here, signed by Manet, the pamphlet also carries the painter's signature in a referential way.The Japanese screen on the left, and the Kuniaki II woodblock, show, in part, from where Manet derived ideas for challenging traditional western perspective. The bold cut-off quality is also, of course, partially attributable to the influence of photography.

When Daguerre and Talbot launched photography in 1839, Paul Delaroche pronounced: "From today, painting is dead!" But while the coincidence of the birth of photography and the crisis of painting is inseparable, and while photography did disrupt the course of painting, it did not extinguish it, indeed it may even have given it new life, opening the doors to radical formal experimentation.
It was not simply that Manet was able to use photographs of Georges Clemenceau in a preparatory capacity when painting the politician's portrait, but being released from mimetic imperatives, when painting his harrowing portrait of Berthe Morisot in mourning, he produced a work of extraordinary viscerality, long before the concept of expressionism was conceived.

The scale of some of the portraits is unexpected. The inconsequential figure, The Street Singer is as large as the important personage of Clemenceau. And it is interesting to see our own George Moore there. Manet did three portraits of Moore around 1879; this one shows the writer in the garden of the artist's studio.

Moore started out in the École des Beaux-Arts, but once he realised the arduous training involved, informed his valet that he would be painting no more. He championed both Zola and Manet, and promoted Manet as the exemplar of the modern.

Manet: Portraying Life is at the Royal Academy of Arts, London until April 14th