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Michael, It seems you have the Learning Paradox (Fodor) also know as the 'Meno paradox' (Plato) in mind. For those not familiar with it: It is impossible to learn something really new because either you know what you are looking for (and then it is not new) or you will never know when you have found it (if you don't know what your looking for). Or in this case: you can never attain a new stage by learning, because that would require expressing the key insights of the new stage in terms of the previous stage (which is impossible by defintion) first as hypothesis, in order to be able to next test its usefulness. However, Piaget would probably not go along with the idea that learning has to be kind of hypothesis testing, and claim that the new stage is formed in the act of reflecting abstraction. I always use the following simple example to illustrate Relecting Abstractions. Consider the jump from addition to multiplication: Adding the number of times you repeat a simple addition amounts to multiplication. ?? Exactly how this insight comes about I don't know, but it seems perfectly possible because it is literaly equivalent. Next, applying the inverse operation (division) seems also to be possible without invoking that which has to be attained before it has been attained. ?? It will will take time and effort and again: exactly how this insight comes about I don't know, but it seems possible because mastering or discovering the inverse operation is what has happend to all operations so far. Next, after some generalization, the fractions will be discovered! ???But this is genuinly something new (a whole new world opens up). And no hypothesis testing is involved!