In short being backed by something means you can convert it into that. So money that is backed by gold can be traded in for an amount of gold. Bitcoins are not backed by math or computer cycles or anything else because they cannot be traded in for it.
So while you can sort of turn electricity or computer cycles into Bitcoins you cannot turn bitcoins into electricity or computer cycles and so you can't say that bitcoins are backed by either.
Edit: it's worth noting that the US dollar is also not backed by anything however.
Not true - the US dollar is backed by US government taxation. The dollar amount you owe in taxes remains unchanged, while the amount of gold or bitcoin you'd need to sell to pay your taxes is determined only by market forces.
If hypothetically everyone in the US were paid in bitcoin, they'd still owe taxes in USD, which would probably lead to a rush on USD around April 14.
Your example is a classic example of exchange rate problems surrounding floating currencies. Since both the USD and bitcoins are free-floating (though there are powerful institutions stabilizing the USD such as the Fed and IMF), there would be an onus to convert USD into Bitcoin until the relative value of USD was high enough to offset the conversion deficit. That, or there would need to be equally powerful institutions behind Bitcoin to buy other currencies, but it would be very hard given that there is a practical limit to the total number of Bitcoins in and out of circulation.
Opposite of fiat is representative currency, which is tied to specific tangible products (frequently previous metals).
Introduction of taxation does not convert a fiat currency into a representative one.
Bitcoins, by the way, can easily be converted into computation and energy. The simplest and most obvious way to do that would be Bitcions->USD->AmazonEC2. Of course there's a billion other ways to convert Bitcoins into useful things, just like gold, because they're simply a store of value with little use for other purposes.
[EDIT] Feel free to refute my argument rather than downvoting me.
This is of course a fuzzy line. And you're right, there are other definitions of "backing" which render it a meaningless term. But the distinction is a useful one for discussing these matters, so what terminology would you prefer instead?
But since you replied I will say: Yes it is backed by nothing. There is no central bank. It's funny that you draw comparisons to gold's lack of intrinsic value because the whole point of central banking is to move away from storing shiny objects in vaults, and in that regard Bitcoin is a regression. (Replace shiny objects with shiny prime numbers.)
And, as I mentioned in a cousin comment, I think the whole definition of backing is just silly. Bitcoin is backed by trust in math and computation, but gold and the USD are backed by trust in people and government respectively. Gold is trusted since it's been in use for thousands of years and is widely used as a store of value. The USD is trusted because the US government has been relatively stable (and powerful) for a while now.
OK, let's suppose for some reason there is a sudden inflation of the US dollar...
Please, asveikau, can you explain a single precise example of what the "central bank" can do to help in this case that causes a net improvement?
Will they invent an "un-printing press"? Will they get people to turn in the excess dollars by exchanging them with gold? Please, asveikau, explain what it is that a central bank can do in this situation that justifies its existence.
However, you are right though that adjusting reserve requirements does generate a net sink for excess monetary supply, though it is of limited capacity. Also, it is not a very viable tool right now, since bank reserves are already at an all time high.
It's backed by math and computational complexity, which is decidedly not "nothing." If, as you say, it is backed by nothing, go ahead and forge some Bitcoins. Except, of course, you cannot forge them. You can only create them via the investment of hardware, electricity, time, and luck.
It's not like gold is actually worth $1,500 an ounce due to its value in making jewelry or in manufacturing. The value of gold is set by the speculative market, just like Bitcoin.
Of course, you're right that it's volatile. But that has nothing to do with its "backing" and everything to do with the fact that 99% of the world has no idea what the hell to make of it yet. That, and the fact that there has never been a decentralized digital currency before, so nobody knows what's going to happen.