The writing process of the paper on the Church-Turing thesis, random processes and platonism has come to an end. The paper is forthcoming in Philosophia Mathematica. A more or less final version of the paper is available here. (wrt to earlier versions, I added a bunch of remarks and answers to possible counterarguments, explained what Church randomness is and why Church-random sequences are not Turing-computable and elaborated on the intuitive source of the conceptual conflation involved).
Many thanks to Leon Horsten, Liesbeth De Mol, Robert Thomas, Jeffrey Ketland, Gualtiero Piccinini, Stanisław Dercz and two anonymous referees for discussion and comments.
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