A strange beast, the nation-state. There was Nick Robinson, the BBC's political editor, reporting from Arsenal's Emirates Stadium where Arsene Wenger and sundry other Frenchmen rule the roost. All smiles, reported the BBC man, as Wenger showed Gordon Brown and Nicolas Sarkozy around.

No-one, said the journalist, was mentioning that "we just lost to them" at the football.

Hold on, Mr Robinson, thought A Viewer. "We"? If memory serves, we beat France home and away, and when it mattered. In point of fact, haven't "we" just earned a very creditable draw against Croatia, the team that really stuffed England? But never mind.

The wholly trivial anecdote is intended only to make a point: in football, as in reality, Europe is complicated. The relationships between its nations and its states cannot be described in shorthand. President Sarkozy's state visit may (at least according to Sarko) herald a new Franco-British entente, or even (according to Mr Brown) an "entente formidable". But what might that mean, precisely, for the numerous interested parties?

A case in point. The President of France arrived keen to press ahead with a nuclear power deal that looks like a good bit of business for his country. Despite the discovery of a large quantity of radioactive debris on the bed of the Rhone, France has never fallen out of love with nuclear's promise of energy security. It leads the world in that technology at a time, fortuitously, when the London government fancies building new stations.

Indeed, according to John Hutton, the Business Secretary, busy this week working on the deal with Jean-Louis Borloo, the French Energy Minister, nuclear could become Britain's "new North Sea". Mr Hutton forgets that the country within Britain with some claim to existing North Sea resources has a government, and a people, disinclined to embrace the nuclear option. President Sarkozy is probably oblivious to the fact.

Gordon Brown is not, however. He does not flinch, these days, on discovering that one of his new best chums is notably right-wing. Nevertheless, the Scottish Prime Minister knows that the idea of a French affair, whatever it might be worth, is a hard sell in parts of Britain. There is an irony in that.

The London press, wondering how Eurosceptic England will "come to terms" with Sarkozy's plea for friendship, forgets that Scots tend to have fewer hang-ups. We have some history with France. We do not, however, desire a nuclear programme. Just to complicate the irony, English opinion, Francophobe or not, is less sanguine about a nuclear future than Downing Street, or the people of France.

Who is dealing with whom, on what terms? Underlying perceptions matter, like it or not. Few of the London tabloids will be hopping on board with the Frogs. Perceptions of what matters, meantime, mean different things in different places. President Sarkozy says that Britain and France "need one another". The view is not widely-held in England. It has not been held, not obviously, not lately, by Mr Brown.

He understands a trade deal. Defence agreements are something else altogether. Mr Sarkozy admires the United States; Mr Brown is a paid-up Atlanticist putting British blood and treasure unquestioningly at America's service: another matter entirely. The French President hints that he will end his country's 42-year estrangement from Nato. But he also persists in talking about an enhanced European defence capability.

Mr Brown (and Washington) will take the former, but run a mile from the latter. All that has actually emerged from the Sarkozy visit in terms of defence, in fact, is a sop to arms trade and domestic budget demands. You can "develop European military capabilities" in a number of ways, but a "joint industrial strategy for complex weapons" sounds like the heart of this matter. Mr Brown will not dare more.

The Prime Minister will also welcome President Sarkozy's free-market rhetoric, but note his economic protectionism, not to mention the fact that the hyper-energetic new man in French politics is having a very hard time persuading his country to accept changed work practices, or "reforms" to their admirable pension and healthcare systems.

The Prime Minister has been wooed, in fact, by a President who wouldn't get a table at a bistro if the decision was left to a popular vote. Sarkozy's fall in esteem has been so sharp it makes Mr Brown's travails seem like a blip. An over-publicised marriage to a singer and model has not helped - the French expect discretion and dignity from their Presidents - but the drubbing the President's party took in recent local elections cannot be blamed entirely on Carla Bruni-Sarkozy. Her husband is sensitive about his height, talks big, but is a diminished political figure.

It is open to question, then, whether Prime Minister Brown or President Sarkozy can attract many revellers to a new Franco-British party. The former is not holding his breath for an outbreak of popular enthusiasm; nor is he likely to hear much encouragement from Washington. Agreements on terrorism or immigration are not contentious (more's the pity). But binding Britain to a relationship with Europe via a cross-Channel pact will trouble the leader of an insular state. In other words, it would not be popular. Mr Brown - the sole reason we stayed outside the euro - knows as much.

For his part, President Sarkozy has delivered some mixed messages this week, albeit with panache. Politically weak at home, he seeks friends abroad. Yet while he calls for "brotherhood" with Britain, he remembers to say that the Franco-German axis remains Europe's "indispensable driving force".

Reportedly, however, he and Germany's Chancellor Angela Merkel do not get on. A state visit to Britain begins to look like a move in the diplomatic game that began long before the EU was imagined. Bonn is perhaps being reminded that Paris can build alliances outside the usual core relationship.

Sarkozy was clever, this week, to point out that Britain is valuable to Europe because of its relationships with America and the Commonwealth. He went on to add, however, that Britain needs Europe if it is to avoid "isolation". A neat formulation, superficially, but, in reality, it illustrates this country's profound ambivalence. The old pretence that we can look in two directions simultaneously continues. This is not an assent, whatever the man from the Elysee would like to pretend.

It is not much comfort, either, for - let's say - a Francophile Scot who thinks in terms of a European future beyond big-state politics. The pomp of the Sarkozy visit and the rhetoric traditional in such events cannot obscure how disparate, how complicated, Europe is. The standard, bland 36-page joint communique as the party drew to a close simply dodged the problem.

Instead, two leaders would like to reform and expand both the G8 and the UN Security Council. Surely Washington, Moscow and Beijing won't mind a bit. Brown and Sarkozy would also like peace in Darfur and human rights in Burma. Who wouldn't, but what about Tibet? They want to build new weapons, meanwhile, and contain terrorism, and go on treating immigration with their accustomed severity. Nice.

Such, for what it is worth, is what they agreed, and not what they appeared to be agreeing. There is a world of difference. Our part of the actual world, the European part, will not be transformed for a while, whoever "we" happen to think we are.