Take the decimal expansion of a pair of irrational numbers which have been mutilated at certain unknown points. You are asked to construct a choice sequence between them subject to some arbitrary constraint. Is the task meaningful? Is it mathematics or nonsense?
Suppose there is a class of cellular automata whose behavior, when observed by some Turing tested entity, reliably and uniquely produces the comment 'you know, I rather think these wee beasties are trying to construct a choice sequence between such and such mutilated numbers! Why on earth would they want to?' And, 'oh look, that little fellow there seems the smartest of the bunch- but hang on, maybe he's hit a wall and now there's that other fellow who seems to be going great guns.'
In a sense, if the above thought experiment isn't utterly incoherent in some way I've failed to spot, what happens to the claim that Mathematics is synthetic in any sense or degree? What happens to intuition?
Oughtn't we to chuck both words in the bin?
Indeed, if cellular automata were the paradigm of Maths or Logical thinking taught to us from Primary School onwards, would we not dismiss as arrant nonsense otherwise common-sense seeming notions like 'the principle of displayability' (from Husserlian phenomenology) meaning that every term or concept used has to be defined by a phenomenon (something observable) determined by it- the like of which, as a methodological principle, every discipline displays to some degree or other?
What's a choice sequence?
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