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Albert Einstein gave us the theory of relativity , explained the photoelectric effect and predicted the existence of gravitational waves . He revolutionized our understanding of space, time and gravity.

In fact, he is so famous for his brilliance that "Einstein" is synonymous with "genius." But being a genius doesn't mean you're infallible. So was Einstein wrong about anything?

"For sure, he was wrong about tons of things," Nicolás Yunes , a theoretical physicist at the University of Illinois Urbana-Champaign, told Live Science. "We remember him for the things he was right about, for the most part, because the things he was right about shocked the scientific world and eventually those ripples affected everyone on Earth."

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For example, one of his most famous predictions was marred by a mathematical mistake. In 1916, Einstein correctly realized that the acceleration of matter could produce ripples in space-time that we now call gravitational waves. But when he and the physicist Nathan Rosen described these waves using mathematics, they ran into a problem: Any solution to the equations of general relativity that would allow for gravitational waves "blew up," Yunes said. "They had these singularities … these divergences. And solutions that diverge and explode like that, they can't be physical representations of reality."

As a result, Einstein changed his mind and decided that gravitational waves couldn't exist. He wrote up his results and submitted them to the journal Physical Review , which had recently begun sending papers to outside experts for peer review. An anonymous reviewer caught an error in Einstein's math, and when Einstein learned about it, he was so furious that he withdrew the paper and submitted it to a different journal.

But the reviewer had spotted a real error: Einstein's mathematical infinities were a coordinate artifact. Much like Earth's lines of longitude seem to converge to a "singularity" at the North Pole even though nothing unusual is happening there on the ground, Einstein's math could have been fixed by using a different set of coordinates.

Without Einstein's knowledge, the reviewer befriended and demonstrated the error to Einstein's assistant, who explained it to Einstein. Einstein corrected the error and republished the paper with the opposite conclusion, showing that gravitational waves do, in fact, exist.

Einstein originally thought that gravitational waves existed, then changed his mind when the math didn't work out. However, an anonymous peer reviewer named Howard Percy Robertson managed to secretly help Einstein fix his mathematical errors. (Image credit: MARK GARLICK/SCIENCE PHOTO LIBRARY via Getty Images) Calculating "impossible infinities"

Remarkably, Einstein stumbled on the same error twice — at least, it seemed that way. When examining the mathematics around black holes , he once again calculated impossible infinities ‪—‬ this time, at the edge of a black hole. And once again, he concluded that this must mean the object couldn't exist.

"Einstein remained skeptical of the existence of black holes," John D. Norton , a professor in the Department of History and Philosophy of Science at the University of Pittsburgh, told Live Science in an email. "He was resolute in his judgment that there would be a singularity in space-time already at the event horizon of a black hole , which is now merely regarded as marking the point of no return for those falling into a black hole."

But this time, Einstein was not convinced that he made a mistake. Norton argues this wasn't just stubbornness; it reflected a philosophical belief in how Einstein thought mathematics and physics should relate to each other.

"He was unmoved by alternative analyses that found his mathematical infinities to be merely artifacts of the particular mathematical methods he preferred," Norton said.

Einstein was skeptical of the existence of black holes (pictured here in a 3D render) because his math showed a breakdown in space-time at a black hole's edge. (Image credit: Cavan Images / Luca Pierro via Getty Images) Einstein and quantum mechanics

Einstein's resistance to quantum mechanics is probably his most famous wrong turn. His central objection was to a phenomenon called quantum entanglement : the idea that two particles can be linked in such a way that measuring one instantly affects the other, no matter the distance between them. In a 1947 letter to his colleague Max Born, he wrote , "I cannot seriously believe in it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance."

Einstein thought this instantaneous phenomenon seemed to violate special relativity , which says that nothing can travel faster than light . As a result, he believed quantum mechanics must be incomplete and that there was some deeper, unknown description of reality that would restore order.

"He died not liking quantum mechanics," Yunes said. It wasn't until 1964, almost a decade after Einstein's death, that John Bell proved entanglement was real.

"Today, a lot of the technology that we have relies on quantum mechanics, and so we know it's correct," Yunes said. "But it's still incompatible with general relativity, with [Einstein's] classical theory."

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"It's possible that general relativity is wrong," Yunes said. "It's also possible that quantum mechanics is not the right description when you're talking about systems that are very strongly gravitating and you're looking at Planck-scale-type dynamics," which are incredibly small scales at which quantum effects dominate. For instance, the center of a black hole, which is both compressed to the quantum scale but also sits inside the most intense gravitational field in the universe, is a place where general relativity predicts a singularity that quantum mechanics has no way to describe.

But many of Einstein's mistakes still moved science forward. "General relativity is a good example," Norton said. "Einstein based it on the idea that he was generalizing the principle of relativity to acceleration and on what he soon called Mach's principle. Neither proved compatible with his final general theory of relativity."

Einstein's mistakes might come as a surprise to others, but they didn't to him. Once, when writing a book with his collaborator Leopold Infeld, Infeld told him he was taking special care with it because Einstein's name would appear on it. Einstein laughed and said , "There are incorrect papers under my name, too."

See how much you know about Albert Einstein with our Einstein quiz!

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• Updates on the shortened stage from Malemort to Ussel

• Get in touch! Email Tom with any and all Tour thoughts

Here is the top ten, ahead of stage nine:

1. Tadej Pogacar (UAE Team Emirates-XRG) 28hrs 49mins 07secs

2. Jonas Vingegaard (Team Visma-Lease a Bike): +2mins 42secs

3. Isaac Del Toro (UAE Team Emirates-XRG): +3mins 27secs

4. Remco Evenepoel (Red Bull-BORA-hansgrohe): +3mins 30secs

5. Juan Ayuso (Lidl-Trek): +3mins 34secs

6. Paul Seixas (Decathlon CMA CGM Team): +3mins 55secs

7. Florian Lipowitz (Red Bull-BORA-hansgrohe): +4mins 00secs

8. Lenny Martinez (Bahrain Victorious): +4mins 21secs

9. Mattias Skjelmose (Lidl-Trek): +4mins 57secs

10. Mathias Vacek (Lidl-Trek): +7mins 10secs

More on the decision to shorten the ninth stage from Jeremy Whittle in Bergerac:

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What does it mean to be patriotic? It depends on a country’s politics.

Pride in one’s country is itself relatively common, according to new results from an international edition of The POLITICO Poll: Majorities of adults across the United States, United Kingdom, Spain, Canada, France and Germany all say they are proud of their country.

But that pride looks different in each country. Expressions of patriotism are shaped by a nation’s history and politics. And in a complex political moment when voters’ relationship with each other, their political parties and their countries continues to shift — and nationalist populist movements continue to rise, claiming patriotism as their own — what it means to be patriotic is evolving.

In America, the flag has become a defining feature of national pride, a symbol increasingly claimed by Donald Trump’s MAGA movement. In Canada, people find pride in their political leaders as they seek to counter Washington’s influence. And in Germany, the country’s Nazi history continues to loom over current debates about nationalism.

This article is part of an ongoing project from POLITICO and Public First , an independent polling company headquartered in London, to measure public opinion across a broad range of policy areas.

You can find new surveys and analysis each month at politico.com/poll .

Have questions or comments? Ideas for future surveys? Email us at poll@politico.com .

The survey, conducted by the independent London-based firm Public First, underscores how cultural and political debates shape subtle differences in patriotism across six major democracies of the West.

Tensions over immigration and culture have roiled Europe and the U.S., with right-wing parties often making national identity central to their political message. The symbols and rituals of patriotism also distinguish countries from one another — from debates over what students are taught in schools and whether they should be required to sing the national anthem, to differences in displays of the flag.

Here’s what stands out for each country.

UNITED STATES

The U.S. rallies around its flag

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Pluralities of adults in other countries — and 49 percent of Americans — say the U.S. has the most patriotic citizens. That’s partly because of its overt displays of patriotism, with Americans historically placing national flags on their lawns and outside restaurants and businesses, and leaders frequently wearing flag paraphernalia.

That embrace of the flag stands out. The U.S. flag has long been a revered symbol in the country, and most Americans (51 percent) say they own the national flag. That’s significantly more than adults in the other countries — for instance, just 22 percent of U.K. adults say they own a Union Jack and just 27 percent in France say they own a flag.

Still, the American flag has become somewhat partisan.

Trump and his MAGA movement have made patriotism and nationalism central to Republican politics, and more than 70 percent of his 2024 voters say they own an American flag. Just 44 percent of those who backed former Vice President Kamala Harris say the same. Trump voters are also more likely (52 percent) than Harris voters (33 percent) to say that displaying a national flag outside your house is considered an act of patriotism.

GERMANY

Many adults think their country should be proud of its history. Germans are the exception.

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Near-majorities of adults in the U.S., the U.K., France, Spain and Canada say their country should be proud of its history. Germany is the outlier, due to its Nazi past.

Just 24 percent of Germans say the country should be proud of its history. A majority — 53 percent — say the country should neither be proud nor ashamed. Still, only 14 percent say Germany should be ashamed, and a 61 percent majority say they feel the country has “done enough to apologize for the bad things it did in the past” — potentially helping explain why most Germans say they feel neutral of the country’s history.

Politicians from the far-right Alternative for Germany (AfD) party have argued that the country should reclaim a sense of national pride, including by reshaping childhood education so it involves “less Hitler,” as one version of a manifesto published by a regional branch of the party put it. Forty-one percent of voters who plan to back AfD in future elections say Germany should be proud of its history — more than double the number of total adults who say the same.

While such ideas were once considered politically fringe in Germany, they have increasingly moved into the mainstream. The AfD is in first place in national polls , in front of German Chancellor Friedrich Merz’s conservatives, and it’s even further ahead in two eastern states in which regional elections are set for September.

CANADA

Everybody is ashamed of their leaders — except for Canada

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Amid a growing anti-incumbency backlash across the West, Canada stands out.

When asked what makes them most ashamed of their country, just 22 percent of Canadians pointed to their leaders — significantly lower than the 40 percent of Americans, 45 percent of Brits, 52 percent of the French, 43 percent of Germans and 50 percent of Spaniards.

There’s a reason for their more positive outlook on the government. Aided by Trump’s trade war and what have become routine threats to turn Canada into the “51st state,” Mark Carney, a rookie politician, managed to revive Canada’s Liberal Party and win office in April 2025 with promises to stand up to Washington while transforming the oil-rich country into an energy superpower. Carney is even popular in Western Canada — traditionally Conservative territory — thanks to pipeline promises and his rollback of Justin Trudeau-era climate policy. A recent Leger poll showed he’s more popular than any other politician in Quebec .

SPAIN

Spanish adults are prouder of where they’re from than other Europeans

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While the majority of adults across the European countries surveyed expressed national pride, Spaniards are on a different level. A 52 percent majority pick Spain as the best place to live, while most adults in the U.K. and France pick other countries over their own.

That showed up when we asked about pride in nationality, too: A strong 76 percent majority of adults say they are proud to be Spanish, compared with the 68 percent who are proud to be British, 71 percent who are proud to be French and 60 percent who are proud to be German.

In Spain, national pride goes beyond its politics — which has been and is still somewhat tumultuous. Spaniards are reckoning with the echoes of Francisco Franco’s decades-long dictatorship, competing pro-independence movements in regions like Catalonia and the Basque Country, and an ongoing deep dissatisfaction with the way their country is being run.

And still, they exhibit a more optimistic sense of national pride than their European counterparts, partly because of the country’s status on the world stage, through its athletic success as well as its reputation as a tourism and culture hub.

FRANCE

The French want students to sing the national anthem

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France’s national anthem, the Marseillaise, is a revered national symbol that dates back to the French Revolution.

Most adults in France, 57 percent, say children should be made to sing the national anthem in school, a view that cuts across partisan lines but is most popular among the far-right National Rally (74 percent) and the centrist Ensemble party (67 percent). Meanwhile, just 42 percent of U.K. adults, 32 percent of German adults and 26 percent in Spain say the same.

More broadly, just like the kneeling protests in the NFL provoked outrage among American conservatives, jeering and booing as a form of protest during the French national anthem at sporting events usually stokes controversy — though in France, the furor is often from across the political spectrum.

UNITED KINGDOM

The U.K. still shares views with Europe, a decade after Brexit

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Ten years after Brexit initiated, an effort to take back control of the U.K., The POLITICO Poll suggests British attitudes toward patriotism remain somewhat pessimistic — and broadly aligned with those elsewhere in Europe.

A 46 percent plurality of adults in the U.K. say “you can’t say you’re proud to be British anymore without being judged,” about on par with the share who say the same in other European countries. Though similar shares of U.K., French and Spanish adults say their countries should be proud of their histories, similar shares of them — and German adults — also say they’re ashamed of their leaders now.

The findings suggest that while Brexit reshaped Britain’s relationship with the European Union, whether it fundamentally changed the country’s sense of national identity is more complex. Other studies have revealed a multi-year decline in U.K. adults taking pride in the country’s achievements.

POLITICO’s Tim Ross, Sue Allan, Joshua Berlinger, James Agelos and Aitor Hernández-Morales contributed to this report.