Building on from the previous post on how complex the space of financial cryptography is, this is an informal list of the many disciplines you might need to grok so as to be able to master the Bitcoin network as a running, living entity.

Loosely organised into classical schools.

Computer Science

Protocols, including peer to peer
databases, state machines
Languages: Code Theory, stack-based languages (script)

Cryptography

Hashing and merkle trees.
Familiarity with pseudonymous authentication, asymmetric cryptography e.g., signing with EC, zero knowledge proofs.
You don’t need encryption, but it’s fun. You might want to read about homomorphic encryption (zkSNARKs and the like) but not needed.

Mathematics

Graph theory, Network graph analysis, big-O
statistics and probability theory including bayesian statistics
Distributions (power, poisson, etc),
Perturbation theory
Percolation theory

Economics

game theory, Nash equilibria, incentives, mechanism design, evolutionary stable stratagies
Risk and finance
Austrian (Hayek, Mises)
Macro - fiat, inflation and deflation, usury, seignorage, history of monetary exchange
Intellectual property, open source software, digital rights management

Miscellaneous?

Business Skills (thanks to readers for pointing this out)
Epidemiology (for SIR/SEIR/SIS propagation) although this might be mathematics
Law (contract & rights, also compliance, property, securities law & regulation)
Accounting (bookkeeping with double entry and triple entry, governance for exotic multisig and so forth)
Psychology and Behavioural economics

Post-amble

A more advanced presentation might list the course / book / level to reach so as to have just enough to be able to contribute to a debate….

FTR, the above isn’t my own work, I’ve plagiarised freely, and I don’t understand them myself. Therefore I am doomed to never Fully Master Bitcoin (™) and as penance, must help build something more understandable ;-) Who is Sisyphus now?

See also: Why Bitcoin Education Matters (needs Flash):