Actually the Constitution is pretty specific about what money is, and FRNs are not Constitutional money.
Art 1, Section 10:
No state shall enter into any treaty, alliance, or confederation; grant letters of marque and reprisal; coin money; emit bills of credit; make anything but gold and silver coin a tender in payment of debts; pass any bill of attainder, ex post facto law, or law impairing the obligation of contracts, or grant any title of nobility.
Oh, and Abraham Lincoln's government gave us paper money.