Ironically, though it is true that there are powerful arguments that a socialist country can't efficiently calculate prices due to nonconvexities in production constraints, the same exact argument also refutes the efficient market hypothesis.
Think of it this way: if you're going to be using computational arguments against central planning ("there is no way to do the planning in a straight mathematical form"), you've got to remember that you're really just talking about an algorithm that computes a set of prices. Markets implement such an algorithm by breaking down a economy into parts that are only weakly coupled, and solving the pricing problem within that particular market. There's no technical reason that a central planner can't do the same. The central planner faces the same issues with nonconvexities as the market does, but you can at least match the same set of prices.
Yeah. It's long seemed to me that the empirical argument against socialism/communism was the flawed implementation of central planning for price-setting in the last century. This seems like a strange rhetorical bent to hear from professionals who work in algorithms and frequently see the short-sighted local minima produced by capitalism.
Market solves a different task - it's calculating prices at the given moment of time. The central planning has to produce a function that will give the price for every moment of the plan's duration (the USSR had 5 year planning term). If communists really knew how to do the later they could have used that knowledge to peacefully take over every capitalist country in the world.
"There's no technical reason that a central planner can't do the same."
As another commenter indicated, this is not necessarily both possible and efficient to do with any known model of a computer. Also, to the extent that market economies provide consumer and elite wants and preferences as inputs, they are morally superior to command economies that provide only the elites' wants and preferences as inputs.
As cscurmudgeon asserted, not indicated. Yelling loudly doesn't an argument make, especially when it's clear you've not bothered to think about the problem.
Consider: if no model of a computer can solve price setting, how can the market? Is the market some kind of hyper-computer? (If you think the answer is yes, I encourage you to publish this work, as it'd instantly upend both economics and computer science.)
The market is effectively a distributed computation. It does this by splitting up the intractable problem of solving prices across an entire economy into multiple loosely coupled optimization problems. This allows for a parallel computation to happen. We call these individual computations "firms" and "organizations" and "individuals." There's no technical reason a command economy can't do the same, unless you have a definition of command economy that amounts to "anything that uses a stupid method to calculate prices."
The latter part of what you say is on-the-mark, but it has nothing at all to do with the computability of prices. Simple tip: just because someone points out a basic error in a critique of command economies doesn't mean the person thinks command economies are great.
Think of it this way: if you're going to be using computational arguments against central planning ("there is no way to do the planning in a straight mathematical form"), you've got to remember that you're really just talking about an algorithm that computes a set of prices. Markets implement such an algorithm by breaking down a economy into parts that are only weakly coupled, and solving the pricing problem within that particular market. There's no technical reason that a central planner can't do the same. The central planner faces the same issues with nonconvexities as the market does, but you can at least match the same set of prices.