According to the website of the Transportation Security Administration, the policy on flying without ID has been changed. If you refuse to provide ID, citing your constitutional rights, you'll be denied boarding. But if you tell them you would show them ID if you could, they'll let you fly.
In short, you can fly without ID. But only if you tell them it's because you've lost your ID. If you tell them that you aren't showing ID because you don't have to show ID, that's a security risk of a different variety.