Here is a mathematical paper on Von Mises’ calculation problem. First a little preamble…
Calculation applies to both centralised and decentralised economies. In a decentralised economy the calculations are performed dynamically and inherently by the actions of the consumers themselves. In a centralised economy the calculations must be performed by central planners. Von Mises and Hayek hypothesised that the latter is impossible. Just to clarify that point, calculation was only thought to be unsolvable for a centrally planned economy.
Please be aware that none of the forms of anarchism actually propose a centrally planned economy. In fact, a command economy could not exist in anarchy for one very obvious reason: the complete absence of centralised government. Identity confused laissez-faire capitalists make two inaccurate claims with regards to this, which are also semi-ironically mutually exclusive to one another:
1) The strawman argument that social anarchism constitutes a form of central government, known in ‘ancap’ parlance as “the collective”, and would therefore seek to implement a command economy. While this is plainly false, even were it actually true, then the linked paper proves that wouldn’t in fact be an issue anyway.
2) That the dynamic decentralised economy advocated by social anarchists would also find calculation problematic, particularly in the absence of currency. No Austrian economist has ever claimed any such thing. What capitalists refer to as the calculation problem concerns itself solely with a command economy of the type implemented by authoritarian socialist governments.
Setting all that to one side, the attached paper examines the mathematical and computational reality of calculation in a centrally planned economy, and evaluates the veracity of claims of deceased Austrian economists from the 1930s, that this would somehow be altogether unachievable.
Mathematics is a priori. Not just claimed to a priori, but actually fundamentally and indisputably a priori. Mathematical science is the scientific study of mathematics. Not only is this falsifiable like all true sciences, but also provable, with proofs that can be both verified. Mathematical proof is therefore factual, and not a matter of opinion. One cannot rationally argue with maths. If something is shown to be mathematically possible, then it just is: 2 + 2 = 4, end of.
Computer theory is a branch of mathematics that deals with computation. In common with the rest of mathematics it too is falsifiable, provable, and can be verified. Computer algorithms proofs can also be mathematically validated, thereby eliminating any scope for errors to persist. Computer theory like all mathematics describes irrefutable facts, which are not open to rational argument. The workings of an algorithm are not somehow a matter for debate.
Economics is a so-called social science, not a precise science. Economics is not falsifiable let alone provable. It is non-verifiable, and cannot be validated. There are no “proofs” in economics; it is entirely a matter of educated opinion, until shown to be erroneous by some other means, such as mathematics.
When economists claim a mathematical or computational basis for something, that claim can be empirically tested, and shown beyond all doubt to be true or false – provided the mathematics to do this have been ‘discovered’. This was not the case for the economic calculation problem back in the 1930s, Mises, Hayak et al. were on safe ground; nobody possessed the means to prove them wrong. Fortunately this is no longer the case, and has not been for quite some time. The linked paper mathematically demolishes the notion that economic calculation is problematically unsolvable for centrally planned economy, such as the Soviet Union. No matter how much some people want that not to be the case, it just is, so they need to get past that. Just be aware that the principles outlined within this paper, can also be readily adapted to prove that calculation does not represent a problem for any dynamic decentralised economy either. Studying the paper should impart a better understanding of why that is.