The name for this is called [Ceremonial Deism](https://en.wikipedia.org/wiki/Ceremonial_deism). The idea is that the phrase is supposed to be so removed from any actual religious meaning that it’s not actually a violation of the Establishments Clause respecting a specific religion over others. From the US Supreme Court in 1984:
> … I would suggest that such practices as the designation of “In God We Trust” as our national motto, or the references to God contained in the Pledge of Allegiance to the flag can best be understood, in Dean Rostow’s apt phrase, as a form a “ceremonial deism,” protected from Establishment Clause scrutiny chiefly because they have lost through rote repetition any significant religious content.
Separation of church and state is more of a doctrine that Pres Jefferson wrote in the early 1800s. It is not part of the constitution. The US Constitution forbids the government from establishing religion or to restrict the free exercise of religion.
The statement “In God we trust” has not been interpreted as violating the establishment clause and the Supreme Court has never ruled specifically on it. Might be unlikely though given that it does not specifically specify any religion but I am no legal scholar.
Separation of Church and State in the American constitution isn’t like French *laicité*, which actively limits the expression of religious values within the public sphere. Rather, it’s that the US government can’t legally privilege or prejudice one religion over another, or religious situations in general vs. comparable secular situations.
For instance, in the 1990s the Supreme Court struck down a city law banning voodoo animal sacrifice but left exemptions for kosher animal slaughter; since it was explicitly singling out religions for punishments and protections, it was deemed unconstitutional. Or a couple years ago, they struck down a rule at Boston City Hall banning groups from flying religious flags when they’d let any other sort of secular group fly a flag there as well.
As for why having “In God We Trust” as a national motto, as others have pointed out, there was a Supreme Court case on the matter. It’s so vague and detached from any particular religious group, not to mention being a wholly symbolic motto rather than a law with concrete effects, that it’s not endorsing a faith or infringing upon people’s religious rights in any way. After all, who’s the God referred to in it, the Abrahamic God? Brahma? Osiris? The Neoplatonic supreme One?
It was added during the cold war scare back in 1956 as a way to unite the US against the godless commies. It was passed by the house and the senate and signed into law by the president and has had large public support so there was no reason for the supreme court to weigh in.
So far any challenges to it have been struck down by state supreme courts as they argue it does not specify one god over others and is more ceremonial in nature rather than religious. There simply isn’t enough support yet to try and push it to the supreme court due to the large population of religious in the US.
In the future this may change as non-religious populations increase.
From the Ninth Circuit Court of Appeals decision in *Aronow v. United States.*:
> It is quite obvious that the national motto and the slogan on coinage and currency ‘In God We Trust’ has nothing whatsoever to do with the establishment of religion. Its use is of patriotic or ceremonial character and bears no true resemblance to a governmental sponsorship of a religious exercise. …It is not easy to discern any religious significance attendant the payment of a bill with coin or currency on which has been imprinted ‘In God We Trust’ or the study of a government publication or document bearing that slogan. In fact, such secular uses of the motto was viewed as sacrilegious and irreverent by President Theodore Roosevelt. Yet Congress has directed such uses. While ‘ceremonial’ and ‘patriotic’ may not be particularly apt words to describe the category of the national motto, it is excluded from First Amendment significance because the motto has no theological or ritualistic impact. As stated by the Congressional report, it has ‘spiritual and psychological value’ and ‘inspirational quality.
In other words, paying with money that says “In God We Trust” is hardly a religious act, and so Congress has not imposed a religion by printing money with that motto. The Establishment Clause of the 1st Amendment only prevents to government from forcing religion. It does not mean that they cannot make allusions (and in this case a generic one) to religion.
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