bitcoin it is worth its weight in gold.
You're being facetious of course
Well, yes and no. An immaterial currency can be issued at will; it is the ultimate "fiat money".
(Before you scream "21 million": that bitcoin limit is "guaranteed" only by fuzzy arguments about a complicated economic game, not "by math", and could be changed if the right players agreed to it. Moreover, any kid can duplicate the amount of bitcoins in existence by creating a hard fork of the blockchain and starting to mine it on his laptotp. Anyone who has bitcoins will gain an equal amount of those "series B" bitcoins, accessible through the same private keys, and could trade them independently of his old bitcoins by duplicating his wallet and downloading the kids's client software. Whether those "series B" bitcoins will get a significant market value is a market(ing) question, not a technical one. And, of course, there are the altcoins.)
We can in principle associate some minimum weight to abstract units that are the result of computation, right?
Yes, with current technology there is in theory a minimum amount of useful energy that needs to be disspated (turned into waste heat) in order to perorm any logical operation. That energy can be expressed as mass by Einstein's equation m = E/c^2. For example, a 10 watt lamp in one second burns 10 joules, which is equivalent to (10 kg m^2/s^2)/(300'000'000 m/s)^2 = 0.00000000000000011 kg, or 0.1 picogram of matter, if I didn't miss some zeros. (Quantum computing may get around this limit somehow, but it is not known whether it will ever be usable for problems like this.)
The theoretical minimum energy cost of computations is quite small, much less than the cost that can be achieved with current technology. But even if we use the actual cost, the mass equivalent of the cost of creating 1 bitcoin will be fairly small. Anyone knows how many joules are tipically consumed today to create 1 valid block (25 BTC)?
However, the cost of creating something in the first place is only an upper bound to its "intrinsic value". There are examples of huge civil engineering works that cost billions to build, were never used, and cost millions to demolish -- that is, had negative value. (The dams built by Saddam Hussein to drain the marshes in Southern Iraq may be one example. The Iridium communications infrastructure may be another.) The computing the nonces of orphaned blocks costs as much as computing those of surviving blocks, but those nonces are worthless (except perhaps to cryptographers, who may have a use for them).
Another upper bound for the "intrinsic value" of something could be the cost of duplicating it. The cost of duplicating a bar of gold is the same as creating the first one. For a banknote, printing an extra copy is much cheaper than printing the first one because all the plates and equipment are reused. For information, the cost of duplication (theoretical and in practice) is extremely small.
It can be argued that duplicating information in the bitcoin system does not produce extra bitcoins. But that is not a physical constraint, it is a property of the whole bitcoin system -- not just the protocols and the algorithms, but also of the internet, and, mainly, how people use and react to those things. So, the allegedly valuable properties of the bitcoin
system -- single spending, authentication, limited supply, irreversibility, global reach, etc. -- do not define the "intrinsic value" of a bitcoin; they contribute only to its market value.
Moreover, no one really knows how hard it is to compute a valid block. The trial-and-error method used by miners is just the most efficient method that we know; but there is no proof or other evidence that there is no better way. There may well exist an agorithm that finds the right nonce for a block header in a few microseconds. Or, worse, finds a block that has the same hash of another block but a different contents, including a specific transaction that is not in the first block. Or finds the private key for any given address. So, the theoretical "mass cost" of computing a valid block may be orders of magnitude lower than the trial-and-error cost.
I already mentioned several of your points (e.g. that mining is based on the "currently best know procedure", not the provably optimal procedure).
And I know of course you're not facetious about the (lack of) intrinsic value of bitcoins, but you probably were facetious when you were asking what their weight is...
Good point about quantum computing possibly getting around the (Landauer) limit though. Well outside my field though, so I can only wonder if it'd simply
the limit, or entirely remove it possibly.
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