Ben Carson is getting a lot of flack for saying that he thinks a Muslim should not be President of the United States, because Islam is not compatible with the Constitution. In the sort of news outlets I read this is treated as a gross bit of bigotry, but I think I agree with him. Islam has a very strong tradition of theocracy. Many Muslims believe that the only just political power is an explicitly religious one, and that the only true law is Sharia, the law of God as interpreted by the mullahs. Theocracy, even in a soft form, is anathema to me and absolutely incompatible with the American system.
Don't get me wrong, I understand that many Muslims don't take the Islamic legal tradition any more seriously than most American Catholics take their own. But Islam as practiced and taught in places like Saudi Arabia, Egypt and Iran is absolutely incompatible with democracy, and indeed with any sort of secular society. Islam has a vast body of traditional law -- Sharia -- built up by centuries of interpreting the Koran and other early Islamic texts. The Islamic tradition teaches that all law should be developed in this way, by judges interpreting religious texts. The system has only a very limited place for a legislature, because the general view is that law is a divine thing, not a human creation; it is our job to find God's law in the relevant texts, not reason our for ourselves what would be best.
Over the past decade I have seen many surveys of Arab and Iranian opinion, and a majority in every poll wants their society and government to be explicitly Islamic. Right there you have a major conflict with the U.S. Constitution, which forbids the "establishment" of religion. Whatever most Americans say about it, we do not live in a Christian country, and this is very, very important to me. You can see from the recent experiences of Iran, Turkey, and Egypt that working out how to have a state that is both Islamic and democratic is very hard. Judaism has some of the same problems, and even in Israel (a state founded by secular socialists) the government assigns a large official role to the Orthodox rabbinate; among other things they decide who is a Jew. Catholicism used to create similar problems; until the 1980s Ireland assigned the church a very large role in education, health care, and other parts of the state, and on matters like marriage and divorce Ireland let the pope dictate their law. But Islam has the biggest and most fully developed system of traditional law, and the most prestigious, and so in general this is a bigger problem in Islam than for the rest of us. In Iran or Saudi Arabia a judge faced with a thorny problem turns to religious texts for the answer and frames his ruling in terms of the Sharia tradition; in the U.S., we forbid our judges from relying on religious texts. There is a real, fundamental, important divide here, the divide created by the Enlightenment. For religious leaders to dictate the law is abhorrent to me, and that is the Islamic tradition.
I know there are Muslim countries, such as Indonesia, where Sharia has limited influence on the law. But I'm sure a mullah from Saudi Arabia or Iran would be quick to tell the Indonesian government that they are bad Muslims. I also suspect that any American Muslim in a position to run for Congress would be a very liberal one, and he or she would have to swear off any interest in Sharia. So an American who is a Muslim in the same sense that Bill Clinton or Barack Obama is a Christian would probably be ok with me. But an orthodox Muslim would have a very hard time convincing me that his or her beliefs can be made to fit with the Constitution or the American political tradition.
To get back to Ben Carson, he is too much of a fundamentalist Christian for me to want him as my president; I don't think I could ever vote for a doctor who doesn't believe in evolution. But to me this only proves that the question he raised is a perfectly legitimate one. The sort of religion a candidate espouses is absolutely relevant to his or her fitness to be President, and the main tradition of Islam creates particular problems.