Translated from french by Grok

Douglas Hofstadter’s sequence is, in a way, a distortion of the Fibonacci sequence. It represents a generic case of the existence of an abstract relationship between the near future and the distant past. In contrast to the mindset fostered by differential equation theory, this sequence exhibits a fractal aspect whose complexity is initially interpreted as indeterminism, since this recursive process appears to behave chaotically. Our goal is to show that this ultimately deterministic and intelligible process constitutes a paradigm of non-uniform reasoning.

One characteristic of what is called intelligent reasoning is the synchronic synthesis of knowledge to solve a given problem. For relatively elementary problems—for example, exercises or quick tests—this characteristic seems more than sufficient to solve them. However, truly difficult problems require the use of diachronic synthesis. Although very costly in terms of memory, this method is indispensable. Indeed, only its power makes it possible not only to overcome encountered difficulties but, above all, to understand the complexity of problems in a global way.

In this context, let’s try to analyze the surprising nature of quickly solving a complex problem. It is obvious that this type of resolution can stem from prior knowledge of a similar problem and its solution. We therefore exclude this case from our study. This choice highlights even more the surprising character of the resolution. Let us then propose a possible explanation for this phenomenon: the rapid resolution appears surprising to the observer of the solver because the former makes an implicit inference, namely the continuity of reasoning in cognitive space. This inference implies for the observer that there is no essential phase change in the solver’s reasoning. Thus, for the observer, the near future of the intellectual path can only depend on the present or at most the recent past.

However, if we now consider a type of problem whose heuristic model corresponds to Douglas Hofstadter’s sequence, it becomes clear that the value sought at a given rank does not depend on values at immediately nearby ranks. In this type of problem, local knowledge proves insufficient; only a diachronic—and in a sense global—synthesis allows determination of the desired value. And it is precisely this method used by the solver that creates the surprise effect in the observer: the solver was not locally fast but globally different, and essentially so.

Thus, our paradigm of non-uniform reasoning explicitly shows that the difference between reasoning based on diachronic synthesis and reasoning that is not is in no way quantitative but fundamentally qualitative. Moreover, when the solver belongs to one of the fundamental categories (see our article: M-classification), this qualitative difference changes in nature in an extreme way, ultimately leading to a notion of incomparability in cognitive space.

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