Haha, good point Thelonius. You tripped me up on my own reasoning. Fiat money is, as you say, also conceptual, and therefore abstract.

But the paper it's written on is real, and I can exchange that piece of paper for goods or services with other people who want a bit of paper.

Ultimately however you are right. Money is only valued because it has a value that is recognised by others. But value is an abstract concept that is no more real than the numbers themselves that are used to visually show its quantity.