I've heard some say that Marine Le Pen was like Trump and by voting against her the French have put the brakes on Trump-like populism.
What are the political similarities between Le Pen and Trump?
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There are major differences between the French and American political culture which reflect into their respective political agendas. I am not sure their minds or political programs fully fall into the definition of "ideologies". I sum it up like this.
Extreme vetting or more controls on Muslims; Note that there were significant attempts by Le Pen supporters like Dieudonné and Alain Soral to appeal to French Muslim voters based on anti-Semitic or anti-Israeli rhetoric (Jean Marie Le Pen is Dieudonné's third child godfather).
Anti-gay wedding; although the n°2 official of the national front is openly gay. This sometimes leads to tensions in a party with a homophobic past.
Relationship with media
Marine Le Pen sometimes goes socialist (35h work week, partial nationalisation of banks, ...), sometimes full liberal (stop the family subsidies, ...) while Trump looks like a fully convinced capitalist (within the borders, as protectionism goes). Note that it wasn't always like this, the old national front was much more liberal economically. To some extent it still is when it campaigns in south eastern France, unlike when it campaigns in northern France, as shown here an article in french that discusses the differences between the socialist north and capitalist south of national front voters.
No gun passion in France, hence no gun passion in Le Pen
French people love their universal health care system. Hence Marine le Pen wants to keep it; but not for foreigners. Unlike Mr Trump's attitude towards health care
Marine Le Pen wants a kind of "national preference" for public employment, public housing, ... (this has been part of the national front's program for a long time, and part of why it is anti-constitutional, it goes against equality before the law). I never heard anything like this from Mr Trump.
There's some significant difference between Trump and Le Pen, due in no small part to the major differences between France and the US
France, like a lot of Europe, is dealing with immigrant problems, most notably ISIS attacks being carried out by some in the growing Muslim population. For better or worse, these attitudes translate into a fear of said groups. So while Trump and Le Pen share a desire to limit immigration, the reasons are vastly different. Trump is focused more on illegal immigration, while Le Pen's is more nationalism, openly aiming to all but stop it
The 2017 National Front manifesto renews its commitment to a massive reduction in legal immigration. Ms Le Pen argues French citizenship should be "either inherited or merited". As for illegal immigrants, they "have no reason to stay in France, these people broke the law the minute they set foot on French soil".
Anti-Eurpoean-Union sentiment - As Brexit proved, there is a growing undercurrent of people who want out of the EU. Le Pen seized on that
Published on Saturday, the document, notably short on macro-economic and practical detail, pledges to take France out of the eurozone and – unless the EU agrees to revert to a loose coalition of nations with neither a single currency nor a border-free area – to hold a referendum on France’s EU membership.
While Trump's talk about free trade sounds similar, the US is not in a similar position
France still holds some pretty broad political opinions. For instance, I read these remarks after the primary, where some pretty extreme groups (like open Communists) got significant votes. Le Pen looks considerably more moderate compared to such positions
Yet, [Communist Party Candidate] Jean-Luc Mélenchon got close to 20 percent of the vote. That’s as many votes as Fillon got. As if this alone isn’t bad enough, I suspect that if he — and not Le Pen — had been the one running against Emmanuel Macron in the May 7 runoff, many fewer people would have crossed party lines to avoid externalism and the calls to cross these lines would have been limited to Fillion and a few others.
Populism is a political doctrine that proposes that the common people are exploited by a privileged elite, and which seeks to resolve this. The underlying ideology of populists can be left, right, or center. Its goal is uniting the uncorrupt and the unsophisticated "little man" against the corrupt dominant elites (usually established politicians) and their camp of followers (usually the rich and influential).
Nationalism is a multidimensional concept reflected in the communal identification with one's nation. It is a political ideology oriented towards gaining and maintaining self-governance, or full sovereignty, over a territory of historical significance to the group (such as its homeland). Nationalism therefore holds that a nation should govern itself, free from unwanted outside interference, and is linked to the concept of self-determination. Nationalism is further oriented towards developing and maintaining a national identity based on shared characteristics such as culture, language, race, religion, political goals or a belief in a common ancestry
The main difference is one of context. A nationalist populist running a major country in Europe is a whole different kettle of fish than one running the USA.
Europe has a fairly recent and bloody history with nationalist populism in the mid 20th Century (eg: Nazism and Fascism). So for a lot of people, seeing signs of it returning is really scary (and this is may be part of why it hasn't in fact managed to do so).
The USA on the other hand doesn't have a lot of history with Nationalist Populist politicians, or at least not a recent one, and not a very bad one*. Probably the best example I can think of were the Jacksonian Democrats, who included multiple presidents in the 19th Century, but nobody in living memory. So while such a person leading a country as powerful as the USA may also be vaguely scary, historically the world has survived the experience. Regardless nobody alive remembers the last time it happened.