Why not? All the Constitution says is that "The Congress shall have power:... To declare war". It is silent on any parameters. The argument, made against the few who wanted a formal Declaration of War post 911, that one wasn't possible because you couldn't define its target, is dangerous. The Constitution is not a suicide pact. If anything, any group, any concept can not only formally declare war on us, but back it up with significantly lethal and damaging acts we must be able to respond. 'War' may not always be the best option, but all options must be on the table.
Not only is the proposed inability dangerous, it is insulting... insulting to the collective imagination of Congress. So you, personally, can't think of a way to define the enemy and make it sound nice. First, declarations of war should NOT sound nice. Second, you, personally, may not know how to define them even un-nicely. I, personally, may not know how to define them. But I, personally, do know how to get them defined! You go to the any Ways and Means Committee and tell them you have an idea for a big tax increase and bipartisan support to enact it, but aren't sure how to phrase the tax bill. I guarantee some staffer will come up with a tax bill targeted to anything you can dream up with significant support from majority and minority committee members once the word bipartisan is mentioned, no matter what the target. Take the bill, change what "tax on" to "war on" and you're ready to go!
Because nations are only equal to nations. Declaring war on an ideology creates a constitutional conflict, because 'ideology' is just another way to say religion. How can government declare war on something the people are supposed to have the freedom of?