Any city worth a flight boasts a memorable edifice: a man-made thing that is its emblem. It’s a picture they will flash up on the news so you know where they’re coming from. London has its clock tower, the gothic rocket attached to the Palace of Westminster, Big Ben. But the home of an ornate grandfather clock is not how Londoners see their city.

The building that is our badge of belonging, the image we’d wear on our sleeve, is a couple of miles to the east. St Paul’s Cathedral is the solemn, eternal boss and hub of our city. We look up for it, mark our bearing by it, judge our distance from the lantern on top. It is the axis of a compass. The great dome is the calm centre of the spinning city. Technically, miles from London may be measured from the cross of Charing, but we know if we were to slog with sledge and snowshoes to reach the pole of London, St Paul’s is where we’d plant our flag and grab the thumbs-up photo.

We look for St Paul’s from the high places on the edge of the metropolis, from the flat marshes down the side of the river, from the Millennium Wheel, from drunken, late-night bus stops and from windy bridges. The dome glows ethereal in the night, London’s own rising moon, the beacon of calm promise. And while London may be the country’s capital, it has also always been its own place. A place apart, a city state within the state. St Paul’s is our cathedral, London’s parish church, born from London fire, risen again like the phoenix, like the resurrection.

That famous photograph of St Paul’s standing like a precious egg against the flame and smoke of the Blitz remains an image of redoubtable fortitude, of London standing against adversity, an image of grit and resistance. Three days after it was printed in Britain, it appeared in German papers, billed as a city in flames, on its knees, London destroyed by the might of the Luftwaffe.

I have shared this city with St Paul’s for one half of one of its three centuries. But I’d never been inside it. As far as I was concerned, it could have been filled with trampolines or angora rabbits: seeing it from the outside is enough Because here’s the thing about St Paul’s: it is the most perfectly beautiful building, the greatest piece of architecture in the capital, and that’s not a contentious, chin-out assertion. It’s plainly, blindingly, unarguably bleeding obvious. Wren is our greatest architect, Shakespeare in stone. St Paul’s is his masterpiece. It is the pre-eminent building in London – and therefore, need we add, in Britain. It is as good as we get.

You approach the west front, with its great door that is open for occasions and looks down to Trafalgar Square. You walk past a statue of a plump, crowned lady on a plinth. Most people assume it’s Victoria, because it’s pretty safe to assume plump ladies on plinths around these parts are Victoria. In fact it’s Queen Anne, the last of the Stuarts, who suffered 17 pregnancies that produced five living babies, only one of which made it past infancy, though the little lad died before she did. Having no heirs, she ushered in the good taste of the Georgians. She was the monarch who saw St Paul’s completed.

You actually get into St Paul’s through a small side door, and are immediately confronted by the sheep-pen nylon ropes, signage and desks of public manipulation. There’s a queue to get a stamp, produce a photo ID, buy popcorn or a ticket. It’s not a great way to enter a great building dedicated to God, and it’s not cheap.

I resent paying for things my ancestors have already bought for me – for more than money. But as the dean, an immensely agreeable chap, points out, you’ve always been charged to get into cathedrals. I expect that’s what they said about burning witches. Making money seems to be part of a cathedral’s point. Moneychangers and temples now happily coexist in an ecumenical, liberal understanding.

I have to shake off the irritation and the disappointment of God’s turnstiles, and take a deep breath to step into the nave of the church. And there it is. A pale, clear, rhythmic, reasoned, ethereal elegance. It is one of the mysteries of human aesthetics that enclosed spaces can elicit such different and extreme feelings in us. A cell doesn’t feel the same as a lift. A supermarket isn’t like an opera house. We can sense minute and subtle changes in the cultural ambience of rooms. Cathedrals, temples, were specifically invented to exploit the drama of enclosed space to make themselves huge parables, echoes of the divine. And as a rule, the inside of a cathedral will be dramatically and intentionally completely different from the outside. That is part of its magic. You walk from the secular world into the spiritual. They are Tardis buildings.

But the initial surprise and the constant shock and the overwhelming joy of St Paul’s is that this inner space is the inverted image of the outer space. That sounds like a simple truism; that the inside of a box feels like the outside of a box. It’s no less a revelation. The outside of the building, based on a stretched Greek cross, looks clean and clear and white. Actually, it’s a carefully harmonious plainsong of decoration: ordered patterns, faces, angles, juxtapositions, interspaced with intense strata of elaborate carvings as beautifully deep, crisp and rhythmic as any in the world. It was Wren’s fortune to be working when stone and woodcarving reached a pitch of excellence in England that was unsurpassed and would never be achieved again, mostly owing to the astonishing genius of the divinely named Grinling Gibbons. Wren insisted that the windows be plain glass, so inside, the decoration is made by the clear light. In the autumn and spring, when the sun is low, the display of chiaroscuro around the pillars and architraves, across the memorials and the black-and-white tiled floor is subtle and sublime. The grades and lines of light and shadow mesmerise as they melt away like cloudfall on dappled downland. St Paul’s was and is the first cathedral of English Protestantism. All the rest were reconsecrated Roman Catholic churches. But here, this room is the first attempt to build a garage, a hangar for a new religion, a new way of walking with the divine, a new spiritual democracy. Honest, egalitarian, open. Not a rich cavern of dark mysteries and unknowable miracles, but a bright, light temple to hard work, personal responsibility and the Word, bright and light enough for every man to read the Word.

Stand under the dome and look up. There, tastefully painted in grisaille, are the authors of the Bible. The three-second echo bounces back the word of the prophets and the apostles from heaven. The word of Wycliffe and Tyndale and Cranmer, the word of the Evangelist, of Christ, of God, of God speaking English. Wren had to encapsulate a theology, ideas and intolerances and schisms, as the country split into a desperate, piecemeal massacre of a civil war that cost a king his head and led the nation to an interregnum of grim guilt and grief. But it also left it shriven, inspired, curious, devout and energetic. And what he built to encapsulate all that was perfectly miraculous. He nailed it first time.

There is no other church in the world that so perfectly conjures the hopes, achievements and central beliefs of Episcopal Protestantism. It is named for the man who took Christianity from being a Jewish sect out to Gentiles and on to the universal. St Paul’s is the hymn of England made stone. So that’s the good bit. Inside the great west door are two easels supporting blown-up posters of orthodox icon faces, those artistically Botoxed gods of the ossified ancient church. In front of them are little metal barbecues of cheap candles, the sort you see in every Catholic church. They are the antithesis of everything this great cathedral was built to inspire – iconoclastic, superstitious bribes of grace. They are anathema and, worse, hideously ugly. The dean, bless him (because someone should), grins and says: “Oh, they’re terribly popular with the public. They really have taken to lighting candles.” Seeing that two-thirds of the visitors are tourists and a good proportion of them will have come from Catholic Europe, this is hardly a surprise. Nor is it the point, particularly in this place.

There are other design problems. A pair of vilely ugly sci-fi paperback-cover paintings of The Agony by a Russian artist are prominently exhibited. In the clear light of Wren’s vision they jerk the head with a physical, philistine slap. Then there is the misshapen and awkward Mother and Child by Henry Moore, which looks like it was hacked out of feta. There is Holman Hunt’s nursery painting The Light of the World, saccharine and sentimental and the best advertisement for atheism I know. These are all displayed with the wall-eyed ease of the Church of England doing relevant, inclusive, contemporary, stop-and-think. You can feel the struggle between high and low church in the extraneous decoration – the distinction between liberal, Anglo-Catholic and Evangelical factions of the Church of England. This is where most of you, understandably, lose interest and start reading the inscriptions on the tombs.

But the biggest aesthetic problem is up the east end. Going past the dome to the choir, with its elaborate stall, misericords and baldachino, and the altar, behind which is a poignant but aesthetically parochial memorial to America’s help in the second world war, it’s topped with a hideous stained-glass window in the apse. And then there are the mosaics. The whole eastern end of the church and the choir, the north and south transepts are eczemaed with Victorian pious pity and hyper-mannerism, turning the clear baroque vision of the cathedral into a cross between a duchess’s jewellery box and the food hall at Harrods. It was Queen Victoria who thought the church too gloomy, too plain.

Despite the ceramic Episcopal graffiti, the space still soars with bright purpose. There’s always been a rivalry between St Paul’s and Westminster Abbey – they refer to each other sniffily as “the other place”. The coronation, of course, happens in the minster, but the cathedral got the big fairy-tale wedding, and the churches get their precedence from the quality of their dead. The minster boasts the nation’s official repository of death, Madame Tussauds in marble: the unknown soldier and Poets’ Corner. But St Paul’s had a coup by snagging Nelson and then Wellington. The Iron Duke’s tomb in the crypt is surrounded by the flags of the nations that fought with him at Waterloo. There is one empty sconce – the Prussian standard was removed in 1914. Nelson is buried in the stone coffin that was carved for Cardinal Wolsey. It had been hanging round the royal family’s attic.

There are a lot of military men down here: Kitchener, who wanted you, lies in his little chapel by the front door – they sing communion just for him once a year; the generals of the American war of independence and the wars against the French. There are memorials to brine-buried sailors, who cluster round Nelson. There’s the great lachrymose Victorian gate, closed for ever on Melbourne, Victoria’s favourite prime minister. Here’s Howard from the Howard League for Penal Reform; here’s Reynolds and, touchingly, across the south transept, just in his eye-line, is the beautiful sculpture of Turner, the Covent Garden barber’s son with a cockney accent and a miraculous eye. Here’s Florence Nightingale and William Howard Russell, the first war correspondent, who made her hospitals possible. Here’s Wycliffe and Philip Sidney, Alexander Fleming, T E Lawrence, Ivor Novello and John Donne, whose ethereal, shrouded tomb figure stands upright beside the choir. He was dean of an earlier incarnation of the cathedral, and is still its presiding muse, the great poet and moral plumb line of London.

Where the abbey and the cathedral really compete is in their music. The cathedral has the better organ and usually the better organist. It also has the horrendous echo and acoustics that can kill noise before it gets to the back wall.

Both churches have choir schools. I sat in as the St Paul’s boys rehearsed some of their anthems for Easter week. They shove and trip into the classroom, noisy and aimless as pinballs, and take that endless, laddie time to settle in the circle of chairs round the piano where their new choirmaster patiently calls for order. They flick through their books and fidget. He says something, hits a note and, without warning, the room fills like an airbag in a crash. It’s replete with noise that seems to push everything to the wall, to fill every space, my ears and my head until the sound leaks down my nose and through the corners of my eyes, a noise that pumps my chest and runs round the pit of my stomach. This noise boys can make is, in the most literal, true, deific sense, miraculous. It’s astonishing not least because it comes out of the spindly, slight, grubby, slouched bodies of urchins, their faces unconcerned that when they open their mouths they can produce the closest thing a mortal can have to superpowers. The choirmaster tells me that a boy’s voice becomes particularly rich and poignant the month before it breaks. There’s something too touching for words in that.

There is a story that Christopher Wren sat up on Parliament Hill to watch the removal of the scaffolding around St Paul’s dome – the second largest in the world after St Peter’s. If it collapsed, as many said it would, he had a carriage waiting to take him to the Channel. It didn’t, but it still must have come as a surprise to the dean and chapter. They thought they were getting a spire. As he sat there and watched the egghead building emerge from the confusion of pulls and pulleys, he might have paused to think that not only was this the first Protestant cathedral of the modern age, but that he was the first architect of cathedrals in the whole history of Christianity to have ever lived to see his creation completed.

While the ashes of the Great Fire of London were still smoking, Wren, along with Evelyn and Hooke, came up with plans for the new city: rational, ordered and reasonable. It never happened because it would have taken a century to build and because Londoners wanted the footprints of their old houses back, so the medieval streets remain, but with every house rebuilt in brick and stone. The fire is the tipping point, the hinge of London’s story. It cauterised the past, the civil wars, the schisms, the religious intolerance, the filth and the romance of the medieval Roman city. With it went the great St Paul’s, a church bigger than the present cathedral, altered and added, decrepit and dangerous. This was where Shakespeare was first published. By the 17th century it was a collapsing, venerable mess that had once boasted a spire taller than Salisbury’s until it was pushed over by a bolt of lightning. Cromwell used it to stable his cavalry.

Wren’s first design for the new cathedral, based on a Greek cross with a dome, was rejected by the dean and chapter – too Roman, too Catholic, too Froggy-foreign. They wanted their old cathedral back, just without the brothels. So Wren, confident in Charles II’s patronage, built a model of another cathedral. The Great Model cost £600, the price of a substantial real house, and it’s still with us. Above the nave is the remarkable, huge and beautiful über-Airfix kit, the greatest matchstick model ever built, the

St Paul’s that never was. Again the church rejected it: too Catholic, too different, domes – no English church had a dome. Steeples were English: straight up, honest, jabbing fingers to God. A cathedral needed a tower. Wren was mortified and furious, and knocked off another drawing, which is also here – a third version of the church. It looks like a hideous, misshapen Cambodian stupa or Las Vegas casino. The dean and chapter, the bishops and the aldermen, the City livery companies – they all loved it, and Wren was given the go-ahead to start building.

The official guide says he made a few alterations over the next 35 years. That’s not quite the truth. It’s a churchy porky. It’s obvious he never intended to build the humpy thing. He knew that this project would last longer than most of the committee, so he never set the plans down on paper. The cathedral that we have, the one he finally built, he kept in his head.

Just stop and consider the magnitude of that for a moment. The only overall plans for St Paul’s Cathedral were in Wren’s head. Look at it. Imagine building it over a generation. It was an unimaginable act of bravado, genius and, some of us would say, faith. He is buried here, of course. He lived to be 91, and his son, also Christopher, wrote his famous epitaph: “Reader, if you seek his monument, look around you.”

Climb the 530 steps to the Golden Gallery on the roof, past the Whispering Gallery. You will see the astounding brilliance of this building. The simplicity is an illusion, as simplicity almost always is. The great geometric complexity of the buttresses that support the immense pressures of the roof is not on the outside of the building, but hidden in the wall. The dome that you see from outside isn’t the one you see from inside when you look up. The one we see from the bus is a thin skin of wood and lead.

The really impossible part of it sits right at the top, the ball and lantern that is the cathedral’s defining image. How could a dome support its weight? It should plummet the 111 metres to the cathedral floor – but it doesn’t, because it’s held up by God and rests on a secret A-frame that squats on the vertical wall at the base of the dome. It’s a brilliant mathematical answer.

St Paul’s isn’t only Wren’s memorial, it is the evocation of brilliant Englishmen: Evelyn and Hooke, the town planner who studied refraction and light and microscopes and made the first drawings of a flea, and Newton, the towering sociopath who said that Wren was the best mathematician of the age.

The final secret of the genius of St Paul’s is that Wren wasn’t originally an architect at all, but an astronomer. Together these men started the Royal Society and launched an age of inquiry into the natural world, the heavens, politics and philosophy. They were all solidly religious, and St Paul’s is the perfect synthesis of the rational and the ethereal, the intuitive and the known.

A moment, a time, a dome of rounded hope and brilliance. When you reach the top of the Golden Gallery and the wind puffs your cheeks and slits your eyes, you squint out over the most spectacular view. When the first tourist stood here in 1710, he would have seen a city that had grown to be the biggest in Europe. Today you look out at what it has become, stretching away on every side, the old river curling down to the sea, and realise there’s something missing. And then you remember – you’re standing on it.