How many people live in Britain? We haven't the foggiest idea

British jobs for British workers. A few weeks ago a young Russian boy started working in our greengrocer's shop in north London, a pleasant, old-fashioned kind of place that somehow survives the competition from Tesco and Waitrose. The Russian has a lovely smile. He came, he told me, from a city in the middle of Russia that made chemicals and there never knew a word of English. Now he speaks it very well. I asked his boss, who's worked in the shop for decades, why he hadn't continued with English boys. "Useless," he said. "They don't wanna know and they can't count. And you have to keep your eye on the till."

In this little street of shops, there are Russians everywhere. Behind the counter in the deli, cutting meat in the backroom of the butcher's, hammering into the tarmac with power drills - not Abramovich Russians but the people the oligarchs fleeced, not EU citizens but here nevertheless and working hard. There are also, of course, Poles, Lithuanians and Latvians as well as the older generation of immigrants who now seem as traditional as half-timbered architecture: Greeks, Turks, Indians, Chinese. On the packed No 19 buses that come up the hill from town, every language in Europe is being talked into mobile phones. Several I can only guess at. It wasn't like this 10 years ago and then suddenly it was. "Nobody told us it would happen" is a weary cliche but true. In fact, at the time of EU enlargement, government politicians said the opposite. People now feel that they have a better understanding of the extent of this social change than their leaders do; that they were there on the front line, seeing and hearing, when the general staff was juggling with numbers back at HQ.

In the science of demography, Britain has returned to the 18th century, when the great riddle and controversy was how many people lived here. People-counting was an ancient idea practised by civilisations and empires that needed numbers to compute taxes and raise armies: the Babylonians, the Chinese, the Romans. King David was trying to count the Hebrews when a plague intervened, thus connecting the idea of census-taking to the wrath of God in much of the Christian world (making it sacrilegious among church-goers and delaying its introduction to the USA): it was a census, after all, that took Joseph back to Bethlehem. At the start of the 18th century, Britain, a country on the brink of world leadership in many branches of science, persisted in using estimates reached by counting the revenues from hearth and window taxes, or from inadequate parish registers and the figures for the consumption of bread. When Jenner successfully experimented with vaccination, just a few years before the century's end, he had no proper estimate of the population to whom it might apply.

The plain method was to choose a day in the year and dispatch enumerators to every house in the country, to record the number of people who had stayed there in the previous 12 hours as well as details of their age and sex. But a bill proposing this vulgar technique was defeated in the Lords in 1753 after loud opposition in the Commons. Mathew Ridley, MP for Newcastle, said that his constituents (Christians, aware of King David's trouble) feared that "some great public misfortune or an epidemical distemper" would follow the numbering. William Thornton, MP for York, protested that a census would be "totally subversive of the last remains of English liberty" (much like identity cards).

The pressure for a census came from two contradictory fears. The first, which ran through most of the century, was that the population was shrinking. This worry was common to most of Europe. Long wars, pandemics and epidemics, the expulsion of the Moors, "decadence" and "effeminacy" among the ruling classes, celibacy in the priesthood: such things combined to produce an idea of failing and unreproductive societies. In Britain, the worry was gin. A Nonconformist minister and insurance broker, the Rev Richard Price, was the foremost proponent of the shrinking theory and calculated in 1780 that the British population had fallen by a nearly a quarter - to 4.5 million - since the Glorious Revolution in 1688. The poor who crowded into the towns, Price's "nurseries of debauchery and voluptuousness", were among the causes. Debauchery, oddly, had not allowed them to breed enough. And then 18 years later the economist Thomas Malthus stood the problem on its head: too many people rather than too few was the danger, the poor were breeding too much. Only in 1801 was informative light shed on the argument. England and Wales contained 9,168,000 people and Scotland 1,600,000.

How many Russians now live in Islington or Lithuanians in Thetford? Even if there were figures, few people would trust them. This week on BBC's Question Time, David Dimbleby asked the audience if they would believe any statistic mentioned by a politician and the audience roared no. The quarrel is no longer between parties - over whether, say, 1.13m of the 2.17m jobs created since 1997 have been filled by foreign nationals (as the government this week eventually conceded) or if that first figure should be 1.5m (as the Tory party claims). Among statisticians themselves there are quarrels, between government departments and between the Office of National Statistics and the Statistics Commission, which was established seven years ago to "help ensure that official statistics are trustworthy and responsive to public needs" by ensuring their quality and integrity. Richard Alldritt, the commission's chief executive, wants the government to spend millions more on improved monitoring of travel movements: the international passenger survey has become a key estimate of migration levels, but Alldritt said it didn't cover every port and that there was "no guarantee that those surveyed give accurate answers and the results have to be scaled up enormously".

How does the survey work? By sampling and extrapolating rather than counting. According to Michael Blastland writing in the Evening Standard, "For ferry passengers, a team in blue blazers stands at the top of each of stairs into the passenger deck and scribbles a quick description of every 10th [passenger] aboard. As the ship sails, the blazers go hunting for their sample, the woman in the green hat, the trucker in overalls by the slot machine, and ask them if they plan to stay, then extrapolate."

We may believe that we are having a new, non-racist and necessary conversation about immigration and population, but the information to hand is the equivalent of counting window tax. The fogs of the 18th century have rolled in again.