Far be it from me to come over all Freudian, but there has to be latent significance in the manifestation of a group of men boring through the ground inside a giant, phallic-shaped machine with a woman’s name. And not just one group of men: we’re talking eight, for eight phallic-shaped machines, all of which bear women’s names.

True, to be fair to the Crossrail project, there are women working these big drilling dicks as well – the evening before I visited the Crossrail site in east London I watched the The Fifteen Billion Pound Railway on BBC2, and the senior engineer working on the particularly tricky job of tunnelling under the Thames was a woman, as was her deputy. However, for now, I suspect these are the exceptions that prove the rule: the grands projets of the built environment have a tendency to be pushed forward by a distinctively masculine combination of grandiosity, obsessive single-mindedness and either brute or mechanical strength.

And, as projects go, they don’t get much grander than Crossrail. I’m not going to bore into you with the statistics here – God knows they’ve been dinned into you enough already; really as noisy background justification for the fact that the vast amount of that £15bn comes out of London taxpayers’ pockets. What exactly are we getting for our buck? Why, big, phallic-shaped machines banging into Mother Earth, that’s what.

Oh, and if you’re not happy with my psychoanalytic take on it, why not hearken to what Terry Morgan, the Crossrail chairman, sold to me as the benefit of the new railway while I was taking my trousers off in front of him. (Purely, I might add, in order that I could put some high-vis ones on before we went down his hole together.) According to Terry – a bluff, genial, highly likable fellow who spent a part of his career in the arms industry – once Crossrail is up and running, it will cut journey times from Reading to the City by 30 whole minutes; not only that, but the harassed commuters won’t have to change trains, they can just sit there playing Angry Birds, or reading the FT the entire time.

I didn’t dance with joy when Terry vouchsafed this – nor, I confess, was I overly impressed by the news that the new Canary Wharf Crossrail station will include three whole subterranean levels chock full of retail outlets. Indeed, I was slightly surprised the chairman was quite so proud of what, on the face of it, went against his earlier statement that “I don’t think of this as an engineering project, it’s building a railway.” I forbore from remarking that it didn’t sound like a railway to me – more like a series of retail opportunities connected by a giant travelator; forbore because, as I say, I liked Terry, and felt rather honoured that he’d decided to accompany me personally as I entered his hole.

Of course, Crossrail’s benefits shouldn’t be anatomised quite so cursorily – there will be new houses, apparently, and millions of ex-urbanites will find themselves within 45 minutes travel time of central London (and its outstanding opportunities to both earn and spend a buck). Overall, London’s public transport infrastructure will receive a 10% boost in capacity – so that’s all right, isn’t it?

Well, it’s all right so long as you agree with a conception of the city that’s embodied in such crude economic metrics. I’ve written for Guardian Cities before about the spatialisation of capital in London’s built environment, and Crossrail represents this just as much as the Shard. There are already three railway lines connecting the western periphery of London with the city centre; the requirement for an entirely new and fast one is a function of one factor alone: the pre-eminence of the financial services industry in the city’s economy, and its ability to get what it wants. Along with high-frequency trading comes the requirement for traders themselves to be transported at high frequencies. Terry Morgan thinks of it as a railway – but I’m more inclined to view it as a colossal cable-router, through which digitised humans will be streamed.

But as I say, I kept my lip buttoned during the security briefing; and stayed shtum as we picked up our oxygen re-breather units, donned hard hats and protective spectacles, and clanked along to the lift that would drop us down into the hole. I didn’t even twit the Crossrail PR flak when he kept on at me about how all the muck that was being scooped out of the ground was being floated downriver and piled up on Wallasea Island, to help construct the RSPB reserve. Nowadays, everyone engaged on grand projects feels the need to cover up the priapism of their endeavours with such environmental fig leafs; but really, I ask you, do we really feel better about being compelled to the daily grind because some Essex birds aren’t getting their feet wet? I think not.

Morgan, quite reasonably, is cock-a-hoop at what his big drilling dicks have done. New laser and hydraulic technology means that the tunnels arrive at the future station locations with margins of a millimetre or less; so precise is their boring, that they were able to pass within inches of St Paul’s. It had taken the Crossrail team a mere eight months to get under the Thames from Plumstead to North Woolwich, whereas poor old Marc Brunel took 16 years.

Instead of workers labouring in unspeakable conditions behind potentially faulty Greathead shields, the Crossrail tunnellers have been safe inside Ada, Phyllis, Victoria, Elizabeth, Mary, Sophia, Ellie and Jessica. Scores of lives were lost tunnelling under London in the past, now safety is paramount. The hole and tunnel, instead of looking – as it would’ve done in Brunel’s day – like something Sebastiao Salgado might want to photograph, is instead an orderly working environment, just like any surface-level construction site, except perhaps a little tidier.

We rode a little train along the tunnel and directly into the tunnelling machine itself. I thought about Thunderbird 2 and the Mole, but all the while Terry and his colleagues kept up a steady stream of information regarding the brilliance of their own equipment, and the efficiency of their labours. I didn’t doubt any of it for a moment – I was even impressed. Sort of.

Great cities are indeed the product of grand projects, among other things; it doesn’t matter how much of an eco-fiend you are, there’s still no disputing the unutterable beauty to be found in the sight of the lights of Los Angeles coming on in the evening smog haze, so that – if you’re standing at the Griffiths Observatory – you see the great reticulation of the streets and freeways flung towards the horizon.

Of course, the greatest of London’s civil engineering projects aren’t quite so visible: Bazalgette’s sewers, the deep-level tube system and now Crossrail. To persist with my Freudianism: arguably, Londoners are the least discontented of the civilised, given that we spend so much of lives contemplating the Tube’s lavatorial tiling, and smelling our fellow citizens’ flatulence.

At the business end of whichever woman-borer we were inside of, we encountered Tom Donahue from Lancaster. Like many of the tunnellers, he’s a former miner – in his case, his early career was marked by mopping-up operations after the Abbeystead disaster of 1984, in which 16 people died when methane gas exploded in an underground water-pumping station. According to Morgan the joy of the work for his team consists in large part simply in the act of tunnelling.

Again: I don’t doubt this – and frankly, given the raw deal miners got from globalisation, I wouldn’t mind if my taxes went to fund lots of their tunnelling. Hell, I wouldn’t care if they bored through the undersides of the city until it resembled nothing so much as a giant emmental cheese. The eight machines used for Crossrail were purpose-built in Germany, and while there’ll be some payback when they’re returned to the manufacturer, basically they’ll soon be pretty much obsolete. Terry mused as to whether or not there might be another use for them, and I suggested they could bore out a large hole in Boris Johnson’s ego. I think Terry smiled at this – but it could have been that the bright sunlight as we regained the surface deceived me.

Terry also wondered aloud what this part of London would look like in 10 years time, as we stood looking over the gallimaufry of plant and the piles of concrete tunnel sections spreading along the bank of Bow Creek. I said I thought it might well be a full-achieved urban scene – but at what cost, given that by then hardly any of the current inhabitants on low incomes will be able to afford to live there.

At this, Terry expostulated: “You’re right there, we’ve got to find a way of providing affordable housing in London.” Ah, yes, we do indeed – and if only it could be done with a bloody great tunnelling machine. But the problem is, it requires a much more intangible and inaccurate piece of equipment: political will.

• Will Self on London’s high-rise future: ‘thrusting, exhilarating, yet strangely insubstantial’