Mathematics and Computation

A blog about mathematics for computers

A comment about “Mathematical undecidability and quantum randomness” by Tomasz Paterek et al.

This is a short note pointing out that the recent paper on“Mathematical undecidability and quantum randomness” by Tomasz Paterek et al. is no black magic, and that the authors are well aware of it. Unfortunately the paper appeared on Slashdot and has since generated an infinite amount of quasi-mathematical discussions.

The paper shows very nicely how to encode provability in a propositional theory into a question about quantum mechanics. It is a cool paper. The authors mention several examples of undecidability, among others also Gödel's result about undecidability of statements in Peano arithmetic. Judging from the garbage that is being generated on the internet, many readers of the paper are jumping to the conclusion that their encoding also applies to axiomatic systems to which Gödel's result applies. But this is not so, and the authors state it clearly in the 4th paragraph on page 1:

“In this paper, we will consider mathematical undecidability in certain axiomatic systems which can be completed and which therefore are not subject to Gödel's incompleteness theorem.”

Ok, did everyone get that? They can only handle finite theories expressed in the propositional calculus. They did not solve an undecidable problem, and they know it.



I'm having trouble seeing the non-trivial content of this paper. Associated to quantum systems there is an obvious notion of physical statements that are definitely true or false, or neither (ie. statements that are, or are not, eigenvectors of some measurement). In an axiom system we have provably true or false, or undecidable. Is it a big surprise that we can encode some simple examples of the latter into the former?

If the Paterek et al. paper interests you, you may find interesting undecidability I have found in the quantum formalism itself. This derives from a logical excluded middle under the Field Axioms and relates to scalars whose logical status are distinct. Some scalars exist as theorems of the Field Axioms, others merely satisfy them. Model Theory proves the undecidability. It then propagates fully throughout a theoremology indicative of causelogy in Nature that explains the "causal anomalies" of Quantum Physics. Some details are in my blog.

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