Bitcoin's Value is Decentralization
Recently I've read a lot of claims that Bitcoins don't have intrinsic value. I've come to a different conclusion. Bitcoins have intrinsic value if they enable desirable interactions that are not possible without them. Bitcoin is a theoretical and practical breakthrough that makes it possible to decentralize services we couldn't previously decentralize.
To elaborate: Bitcoin isn't just a currency but an elegant universal solution to the Byzantine Generals' Problem, one of the core problems of reaching consensus in Distributed Systems. Until recently it was thought to not be practically solvable at all, much less on a global scale. Irrespective of its currency aspects, many experts believe Bitcoin is brilliant in that it technically made possible what was previously thought impossible.
The Byzantine Generals' Problem roughly goes as follows: N Generals have their armies camped outside a city they want to invade. They know their numbers are strong enough that if at least 1/2 of them attack at the same time they'll be victorious. But if they don't coordinate the time of attack, they'll be spread too thin and all die. They also suspect that some of the Generals might be disloyal and send fake messages. Since they can only communicate by messenger, they have no means to verify the authenticity of a message. How can such a large group reach consensus on the time of attack without trust or a central authority, especially when faced with adversaries intent on confusing them?
Bitcoin's solution is this: All of the Generals start working on a mathematical problem that statistically should take 10 minutes to solve if all of them worked on it. Once one of them finds the solution, she broadcasts that solution to all the other Generals. Everyone then proceeds to extending that solution - which again should take another ten minutes. Every General always starts working on extending the longest solution he's seen. After a solution has been extended 12 times, every General can be certain that no attacker controlling less than half the computational resources could have created another chain of similar length. The existence of the 12-block chain is proof that a majority of them has participated in its creation. We call this a proof-of-work scheme.
If that sounds confusing, don't worry. What it means is just that consensus is reached, because computational resources are scarce. You vote with work. To rig the vote an attacker would need to control more computational power than the honest nodes. To ensure it's more expensive for an attacker to purchase the computational power needed to attack the system, Bitcoin adds an incentive scheme. Users who contribute computational power get rewarded for their work. If the value of a Bitcoin rises and thus attacking the system becomes more profitable, it also becomes more profitable for honest users to add computational resources. At any given point, one would expect miners to invest as much resources into mining as is profitable for them. Bitcoin is a currency, because it needs incentives to protect the consensus process from attackers. This computational process ("mining") is not wasteful at all, but an incredibly efficient way to make attacks economically unprofitable. Bitcoin never uses more computational resources than neccessary to protect the integrity of its interactions.
Now let's go back to discussing the value provided by Bitcoin. Essentially it's a means to make consensus in highly distributed large-scale systems, which would otherwise never be able to reach consensus. The value of this is, that it's now possible to build applications in a decentralized fashion, that we previously thought could not be built without a central authority. The most obvious value of Bitcoin is as a medium of exchange for goods and services that can't be easily bought or sold using cash issued by a central authority. But there's more. Bitcoin also makes it possible to fully decentralize the DNS (Domain Name System). In that case, every Bitcoin Domain Name already comes with a cryptographic key pair. That means it also allows us to solve the PKI (Public Key Infrastructure) problem - every name you connect to has an encryption key associated with it that can be verified without trusting a central authority. In case network traffic monitoring prevents people from accessing information either at all or anonymously, Bitcoin makes it feasible to pay for internet relays that anonymize or reroute traffic - that is, it makes it easier to remove central control and fight censorship. The list goes on, I've been hard-pressed to find any decentralization schemes that would not benefit from Bitcoin integration
The pattern here is: Pretty much all of these applications can already be built in centralized form. But often the centralized solution comes with a whole bunch of weaknesses. In the PKI case you'll have to trust over 300 Certificate Authorities every time you make an https:// connection, many of which are located in countries with repressive governments. If any of those 300 entities get compromised or malevolent attackers will be able to read your email, access your bank account, and violate your privacy. DNS is getting censored by governments because they can. And every time you store value in currency, you're trusting a central authority that it isn't mismanaged and thus depreciates in value.
Is there value in Bitcoin? Let me ask a counter question: Is decentralization valuable? If you think that we'll increasingly lose trust in the central authorities that manage the infrastructure we rely on, you might expect Bitcoins to rise a lot in value. If not, that is you believe that authorities will be able to tackle the challenges of the future better in centralized form, then from your perspective Bitcoins don't add value. We'll see.